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class="vector-toc-list"> </ul> </li> <li id="toc-Affine_connections" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Affine_connections"> <div class="vector-toc-text"> 1.2 Affine connections </div> </a> <ul id="toc-Affine_connections-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Klein_geometries_as_model_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Klein_geometries_as_model_spaces"> <div class="vector-toc-text"> 1.3 Klein geometries as model spaces </div> </a> <ul id="toc-Klein_geometries_as_model_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Pseudogroups" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pseudogroups"> <div class="vector-toc-text"> 2 Pseudogroups </div> </a> <ul id="toc-Pseudogroups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formal_definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formal_definition"> <div class="vector-toc-text"> 3 Formal definition </div> </a> <button aria-controls="toc-Formal_definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Formal definition subsection </button> <ul id="toc-Formal_definition-sublist" class="vector-toc-list"> <li id="toc-Definition_via_gauge_transitions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition_via_gauge_transitions"> <div class="vector-toc-text"> 3.1 Definition via gauge transitions </div> </a> <ul id="toc-Definition_via_gauge_transitions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition_via_absolute_parallelism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition_via_absolute_parallelism"> <div class="vector-toc-text"> 3.2 Definition via absolute parallelism </div> </a> <ul id="toc-Definition_via_absolute_parallelism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_principal_connections" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_principal_connections"> <div class="vector-toc-text"> 3.3 As principal connections </div> </a> <ul id="toc-As_principal_connections-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition_by_an_Ehresmann_connection" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition_by_an_Ehresmann_connection"> <div class="vector-toc-text"> 3.4 Definition by an Ehresmann connection </div> </a> <ul id="toc-Definition_by_an_Ehresmann_connection-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Special_Cartan_connections" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Special_Cartan_connections"> <div class="vector-toc-text"> 4 Special Cartan connections </div> </a> <button aria-controls="toc-Special_Cartan_connections-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Special Cartan connections subsection </button> <ul id="toc-Special_Cartan_connections-sublist" class="vector-toc-list"> <li id="toc-Reductive_Cartan_connections" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reductive_Cartan_connections"> <div class="vector-toc-text"> 4.1 Reductive Cartan connections </div> </a> <ul id="toc-Reductive_Cartan_connections-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parabolic_Cartan_connections" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Parabolic_Cartan_connections"> <div class="vector-toc-text"> 4.2 Parabolic Cartan connections </div> </a> <ul id="toc-Parabolic_Cartan_connections-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Associated_differential_operators" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Associated_differential_operators"> <div class="vector-toc-text"> 5 Associated differential operators </div> </a> <button aria-controls="toc-Associated_differential_operators-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Associated differential operators subsection </button> <ul id="toc-Associated_differential_operators-sublist" class="vector-toc-list"> <li id="toc-Covariant_differentiation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Covariant_differentiation"> <div class="vector-toc-text"> 5.1 Covariant differentiation </div> </a> <ul id="toc-Covariant_differentiation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_fundamental_or_universal_derivative" class="vector-toc-list-item vector-toc-level-2"> <a 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data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Generalization of affine connections</div> In the mathematical field of <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, a Cartan connection is a flexible generalization of the notion of an <a href="/wiki/Affine_connection" title="Affine connection">affine connection</a>. It may also be regarded as a specialization of the general concept of a <a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">principal connection</a>, in which the geometry of the <a href="/wiki/Principal_bundle" title="Principal bundle">principal bundle</a> is tied to the geometry of the base manifold using a <a href="/wiki/Solder_form" title="Solder form">solder form</a>. Cartan connections describe the geometry of manifolds modelled on <a href="/wiki/Homogeneous_space" title="Homogeneous space">homogeneous spaces</a>. The theory of Cartan connections was developed by <a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a>, as part of (and a way of formulating) his <a href="/wiki/Method_of_moving_frames" class="mw-redirect" title="Method of moving frames">method of moving frames</a> (repère mobile).<a href="#cite_note-1">[1]</a> The main idea is to develop a suitable notion of the <a href="/wiki/Connection_form" title="Connection form">connection forms</a> and <a href="/wiki/Curvature" title="Curvature">curvature</a> using moving frames adapted to the particular geometrical problem at hand. In relativity or Riemannian geometry, <a href="/wiki/Orthonormal_frame" title="Orthonormal frame">orthonormal frames</a> are used to obtain a description of the <a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a> as a Cartan connection. For Lie groups, <a href="/wiki/Maurer%E2%80%93Cartan_form" title="Maurer–Cartan form">Maurer–Cartan frames</a> are used to view the <a href="/wiki/Maurer%E2%80%93Cartan_form" title="Maurer–Cartan form">Maurer–Cartan form</a> of the group as a Cartan connection. Cartan reformulated the differential geometry of (<a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo</a>) <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>, as well as the differential geometry of <a href="/wiki/Manifold" title="Manifold">manifolds</a> equipped with some non-metric structure, including <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> and <a href="/wiki/Homogeneous_space" title="Homogeneous space">homogeneous spaces</a>. The term 'Cartan connection' most often refers to Cartan's formulation of a (pseudo-)Riemannian, <a href="/wiki/Affine_connection" title="Affine connection">affine</a>, <a href="/wiki/Projective_connection" title="Projective connection">projective</a>, or <a href="/wiki/Conformal_connection" title="Conformal connection">conformal connection</a>. Although these are the most commonly used Cartan connections, they are special cases of a more general concept. Cartan's approach seems at first to be coordinate dependent because of the choice of frames it involves. However, it is not, and the notion can be described precisely using the language of principal bundles. Cartan connections induce covariant derivatives and other differential operators on certain associated bundles, hence a notion of parallel transport. They have many applications in geometry and physics: see the <a href="/wiki/Method_of_moving_frames" class="mw-redirect" title="Method of moving frames">method of moving frames</a>, <a href="/wiki/Cartan_formalism_(physics)" class="mw-redirect" title="Cartan formalism (physics)">Cartan formalism</a> and <a href="/wiki/Einstein%E2%80%93Cartan_theory" title="Einstein–Cartan theory">Einstein–Cartan theory</a> for some examples. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=1" title="Edit section: Introduction">edit</a>]</div> At its roots, geometry consists of a notion of congruence between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a <a href="/wiki/Lie_group" title="Lie group">Lie group</a> on space. Lie groups generally act quite rigidly, and so a Cartan geometry is a generalization of this notion of congruence to allow for <a href="/wiki/Curvature" title="Curvature">curvature</a> to be present. The flat Cartan geometries—those with zero curvature—are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein. A <a href="/wiki/Klein_geometry" title="Klein geometry">Klein geometry</a> consists of a Lie group G together with a Lie subgroup H of G. Together G and H determine a <a href="/wiki/Homogeneous_space" title="Homogeneous space">homogeneous space</a> G/H, on which the group G acts by left-translation. Klein's aim was then to study objects living on the homogeneous space which were congruent by the action of G. A Cartan geometry extends the notion of a Klein geometry by attaching to each point of a <a href="/wiki/Manifold" title="Manifold">manifold</a> a copy of a Klein geometry, and to regard this copy as tangent to the manifold. Thus the geometry of the manifold is infinitesimally identical to that of the Klein geometry, but globally can be quite different. In particular, Cartan geometries no longer have a well-defined action of G on them. However, a Cartan connection supplies a way of connecting the infinitesimal model spaces within the manifold by means of <a href="/wiki/Parallel_transport" title="Parallel transport">parallel transport</a>. <div class="mw-heading mw-heading3"><h3 id="Motivation">Motivation</h3>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=2" title="Edit section: Motivation">edit</a>]</div> Consider a smooth surface S in 3-dimensional Euclidean space R3. Near to any point, S can be approximated by its tangent plane at that point, which is an <a href="/wiki/Affine_subspace" class="mw-redirect" title="Affine subspace">affine subspace</a> of Euclidean space. The affine subspaces are model surfaces—they are the simplest surfaces in R3, and are homogeneous under the Euclidean group of the plane, hence they are Klein geometries in the sense of <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a>'s <a href="/wiki/Erlangen_programme" class="mw-redirect" title="Erlangen programme">Erlangen programme</a>. Every smooth surface S has a unique affine plane tangent to it at each point. The family of all such planes in R3, one attached to each point of S, is called the congruence of tangent planes. A tangent plane can be "rolled" along S, and as it does so the point of contact traces out a curve on S. Conversely, given a curve on S, the tangent plane can be rolled along that curve. This provides a way to identify the tangent planes at different points along the curve by affine (in fact Euclidean) transformations, and is an example of a Cartan connection called an <a href="/wiki/Affine_connection" title="Affine connection">affine connection</a>. Another example is obtained by replacing the planes, as model surfaces, by spheres, which are homogeneous under the Möbius group of conformal transformations. There is no longer a unique sphere tangent to a smooth surface S at each point, since the radius of the sphere is undetermined. This can be fixed by supposing that the sphere has the same <a href="/wiki/Mean_curvature" title="Mean curvature">mean curvature</a> as S at the point of contact. Such spheres can again be rolled along curves on S, and this equips S with another type of Cartan connection called a <a href="/wiki/Conformal_connection" title="Conformal connection">conformal connection</a>. Differential geometers in the late 19th and early 20th centuries were very interested in using model families such as planes or spheres to describe the geometry of surfaces. A family of model spaces attached to each point of a surface S is called a congruence: in the previous examples there is a canonical choice of such a congruence. A Cartan connection provides an identification between the model spaces in the congruence along any curve in S. An important feature of these identifications is that the point of contact of the model space with S always moves with the curve. This generic condition is characteristic of Cartan connections. In the modern treatment of affine connections, the point of contact is viewed as the origin in the tangent plane (which is then a vector space), and the movement of the origin is corrected by a translation, and so Cartan connections are not needed. However, there is no canonical way to do this in general: in particular for the conformal connection of a sphere congruence, it is not possible to separate the motion of the point of contact from the rest of the motion in a natural way. In both of these examples the model space is a homogeneous space G/H. <ul><li>In the first case, G/H is the affine plane, with G = Aff(R2) the <a href="/wiki/Affine_group" title="Affine group">affine group</a> of the plane, and H = GL(2) the corresponding general linear group.</li> <li>In the second case, G/H is the conformal (or <a href="/wiki/Celestial_sphere" title="Celestial sphere">celestial</a>) sphere, with G = O+(3,1) the <a href="/wiki/Lorentz_group" title="Lorentz group">(orthochronous) Lorentz group</a>, and H the <a href="/wiki/Group_action_(mathematics)#Orbits_and_stabilizers" class="mw-redirect" title="Group action (mathematics)">stabilizer</a> of a null line in R3,1.</li></ul> The Cartan geometry of S consists of a copy of the model space G/H at each point of S (with a marked point of contact) together with a notion of "parallel transport" along curves which identifies these copies using elements of G. This notion of parallel transport is generic in the intuitive sense that the point of contact always moves along the curve. In general, let G be a group with a subgroup H, and M a manifold of the same dimension as G/H. Then, roughly speaking, a Cartan connection on M is a G-connection which is generic with respect to a reduction to H. <div class="mw-heading mw-heading3"><h3 id="Affine_connections">Affine connections</h3>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=3" title="Edit section: Affine connections">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Affine_connection" title="Affine connection">Affine connection</a></div> An <a href="/wiki/Affine_connection" title="Affine connection">affine connection</a> on a manifold M is a <a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">connection</a> on the <a href="/wiki/Frame_bundle" title="Frame bundle">frame bundle (principal bundle)</a> of M (or equivalently, a <a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">connection</a> on the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle (vector bundle)</a> of M). A key aspect of the Cartan connection point of view is to elaborate this notion in the context of <a href="/wiki/Principal_bundle" title="Principal bundle">principal bundles</a> (which could be called the "general or abstract theory of frames"). Let H be a <a href="/wiki/Lie_group" title="Lie group">Lie group</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}"> its <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a>. Then a principal H-bundle is a <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundle</a> P over M with a smooth <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">action</a> of H on P which is free and transitive on the fibers. Thus P is a smooth manifold with a smooth map π: P → M which looks locally like the <a href="/wiki/Trivial_bundle" class="mw-redirect" title="Trivial bundle">trivial bundle</a> M × H → M. The frame bundle of M is a principal GL(n)-bundle, while if M is a <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>, then the <a href="/wiki/Orthonormal_frame_bundle" class="mw-redirect" title="Orthonormal frame bundle">orthonormal frame bundle</a> is a principal O(n)-bundle. Let Rh denote the (right) action of h ∈ H on P. The derivative of this action defines a <a href="/wiki/Vertical_bundle" class="mw-redirect" title="Vertical bundle">vertical vector</a> field on P for each element ξ of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}">: if h(t) is a 1-parameter subgroup with h(0)=e (the identity element) and h '(0)=ξ, then the corresponding vertical vector field is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{\xi }={\frac {\mathrm {d} }{\mathrm {d} t}}R_{h(t)}{\biggr |}_{t=0}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{\xi }={\frac {\mathrm {d} }{\mathrm {d} t}}R_{h(t)}{\biggr |}_{t=0}.\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3632d44fa65c51367b8a7d5b8e6d3048147d07f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.375ex; height:5.676ex;" alt="{\displaystyle X_{\xi }={\frac {\mathrm {d} }{\mathrm {d} t}}R_{h(t)}{\biggr |}_{t=0}.\,}"></dd></dl> A principal H-connection on P is a <a href="/wiki/Differential_1-form" class="mw-redirect" title="Differential 1-form">1-form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \colon TP\to {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>:</mo> <mi>T</mi> <mi>P</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \colon TP\to {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e14e633c91a647ebabdcbe5879217b65a5f0ef73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.687ex; height:2.509ex;" alt="{\displaystyle \omega \colon TP\to {\mathfrak {h}}}"> on P, with values in the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}"> of H, such that <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hbox{Ad}}(h)(R_{h}^{*}\omega )=\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Ad</mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hbox{Ad}}(h)(R_{h}^{*}\omega )=\omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/611aa0b6a81326e183fbedb39f65361004465ad3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.926ex; height:3.009ex;" alt="{\displaystyle {\hbox{Ad}}(h)(R_{h}^{*}\omega )=\omega }"></li> <li>for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi \in {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36cd53281a600b622ce02d2e1c2c78ac2fe59cad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.082ex; height:2.509ex;" alt="{\displaystyle \xi \in {\mathfrak {h}}}">, ω(Xξ) = ξ (identically on P).</li></ol> The intuitive idea is that ω(X) provides a vertical component of X, using the isomorphism of the fibers of π with H to identify vertical vectors with elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}">. Frame bundles have additional structure called the <a href="/wiki/Solder_form" title="Solder form">solder form</a>, which can be used to extend a principal connection on P to a trivialization of the tangent bundle of P called an absolute parallelism. In general, suppose that M has dimension n and H acts on Rn (this could be any n-dimensional real vector space). A solder form on a principal H-bundle P over M is an Rn-valued 1-form θ: TP → Rn which is horizontal and equivariant so that it induces a <a href="/wiki/Bundle_homomorphism" class="mw-redirect" title="Bundle homomorphism">bundle homomorphism</a> from TM to the <a href="/wiki/Associated_bundle" title="Associated bundle">associated bundle</a> P ×H Rn. This is furthermore required to be a bundle isomorphism. Frame bundles have a (canonical or tautological) solder form which sends a tangent vector X ∈ TpP to the coordinates of dπp(X) ∈ Tπ(p)M with respect to the frame p. The pair (ω, θ) (a principal connection and a solder form) defines a 1-form η on P, with values in the Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}"> of the <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a> G of H with Rn, which provides an isomorphism of each tangent space TpP with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}">. It induces a principal connection α on the associated principal G-bundle P ×H G. This is a Cartan connection. Cartan connections generalize affine connections in two ways. <ul><li>The action of H on Rn need not be effective. This allows, for example, the theory to include spin connections, in which H is the <a href="/wiki/Spin_group" title="Spin group">spin group</a> Spin(n) rather than the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> O(n).</li> <li>The group G need not be a semidirect product of H with Rn.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Klein_geometries_as_model_spaces">Klein geometries as model spaces</h3>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=4" title="Edit section: Klein geometries as model spaces">edit</a>]</div> Klein's <a href="/wiki/Erlangen_programme" class="mw-redirect" title="Erlangen programme">Erlangen programme</a> suggested that geometry could be regarded as a study of <a href="/wiki/Homogeneous_space" title="Homogeneous space">homogeneous spaces</a>: in particular, it is the study of the many geometries of interest to geometers of 19th century (and earlier). A Klein geometry consisted of a space, along with a law for motion within the space (analogous to the <a href="/wiki/Euclidean_transformation" class="mw-redirect" title="Euclidean transformation">Euclidean transformations</a> of classical <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>) expressed as a <a href="/wiki/Lie_group" title="Lie group">Lie group</a> of <a href="/wiki/Transformation_group" class="mw-redirect" title="Transformation group">transformations</a>. These generalized spaces turn out to be homogeneous <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifolds</a> diffeomorphic to the <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">quotient space</a> of a Lie group by a <a href="/wiki/Lie_subgroup" class="mw-redirect" title="Lie subgroup">Lie subgroup</a>. The extra differential structure that these homogeneous spaces possess allows one to study and generalize their geometry using calculus. The general approach of Cartan is to begin with such a smooth Klein geometry, given by a Lie group G and a Lie subgroup H, with associated Lie algebras <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}">, respectively. Let P be the underlying <a href="/wiki/Principal_homogeneous_space" title="Principal homogeneous space">principal homogeneous space</a> of G. A Klein geometry is the homogeneous space given by the quotient P/H of P by the right action of H. There is a right H-action on the fibres of the canonical projection <dl><dd>π: P → P/H</dd></dl> given by Rhg = gh. Moreover, each <a href="/wiki/Fibre_bundle" class="mw-redirect" title="Fibre bundle">fibre</a> of π is a copy of H. P has the structure of a <a href="/wiki/Principal_bundle" title="Principal bundle">principal H-bundle</a> over P/H.<a href="#cite_note-2">[2]</a> A vector field X on P is vertical if dπ(X) = 0. Any ξ ∈ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}"> gives rise to a canonical vertical vector field Xξ by taking the derivative of the right action of the 1-parameter subgroup of H associated to ξ. The <a href="/wiki/Maurer-Cartan_form" class="mw-redirect" title="Maurer-Cartan form">Maurer-Cartan form</a> η of P is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}"><a href="/wiki/Lie_algebra-valued_form" class="mw-redirect" title="Lie algebra-valued form">-valued one-form</a> on P which identifies each tangent space with the Lie algebra. It has the following properties: <ol><li>Ad(h) Rh*η = η for all h in H</li> <li>η(Xξ) = ξ for all ξ in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}"></li> <li>for all g∈P, η restricts a linear isomorphism of TgP with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}"> (η is an absolute parallelism on P).</li></ol> In addition to these properties, η satisfies the structure (or structural) equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\eta +{\tfrac {1}{2}}[\eta ,\eta ]=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>η</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">[</mo> <mi>η</mi> <mo>,</mo> <mi>η</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\eta +{\tfrac {1}{2}}[\eta ,\eta ]=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e483a00d4008d34cea1fa0fb45c416ce0b153342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.458ex; height:3.509ex;" alt="{\displaystyle d\eta +{\tfrac {1}{2}}[\eta ,\eta ]=0.}"></dd></dl> Conversely, one can show that given a manifold M and a principal H-bundle P over M, and a 1-form η with these properties, then P is locally isomorphic as an H-bundle to the principal homogeneous bundle G→G/H. The structure equation is the <a href="/wiki/Integrability_condition" class="mw-redirect" title="Integrability condition">integrability condition</a> for the existence of such a local isomorphism. A Cartan geometry is a generalization of a smooth Klein geometry, in which the structure equation is not assumed, but is instead used to define a notion of <a href="/wiki/Curvature" title="Curvature">curvature</a>. Thus the Klein geometries are said to be the flat models for Cartan geometries.<a href="#cite_note-3">[3]</a> <div class="mw-heading mw-heading2"><h2 id="Pseudogroups">Pseudogroups</h2>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=5" title="Edit section: Pseudogroups">edit</a>]</div> Cartan connections are closely related to <a href="/wiki/Pseudogroup" title="Pseudogroup">pseudogroup</a> structures on a manifold. Each is thought of as modelled on a Klein geometry G/H, in a manner similar to the way in which <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a> is modelled on <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>. On a manifold M, one imagines attaching to each point of M a copy of the model space G/H. The symmetry of the model space is then built into the Cartan geometry or pseudogroup structure by positing that the model spaces of nearby points are related by a transformation in G. The fundamental difference between a Cartan geometry and pseudogroup geometry is that the symmetry for a Cartan geometry relates infinitesimally close points by an infinitesimal transformation in G (i.e., an element of the Lie algebra of G) and the analogous notion of symmetry for a pseudogroup structure applies for points that are physically separated within the manifold. The process of attaching spaces to points, and the attendant symmetries, can be concretely realized by using special <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate systems</a>.<a href="#cite_note-4">[4]</a> To each point p ∈ M, a <a href="/wiki/Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighborhood</a> Up of p is given along with a mapping φp : Up → G/H. In this way, the model space is attached to each point of M by realizing M locally at each point as an open subset of G/H. We think of this as a family of coordinate systems on M, parametrized by the points of M. Two such parametrized coordinate systems φ and φ′ are H-related if there is an element hp ∈ H, parametrized by p, such that <dl><dd>φ′p = hp φp.<a href="#cite_note-5">[5]</a></dd></dl> This freedom corresponds roughly to the physicists' notion of a <a href="/wiki/Gauge_fixing" title="Gauge fixing">gauge</a>. Nearby points are related by joining them with a curve. Suppose that p and p′ are two points in M joined by a curve pt. Then pt supplies a notion of transport of the model space along the curve.<a href="#cite_note-6">[6]</a> Let τt : G/H → G/H be the (locally defined) composite map <dl><dd>τt = φpt o φp0−1.</dd></dl> Intuitively, τt is the transport map. A pseudogroup structure requires that τt be a symmetry of the model space for each t: τt ∈ G. A Cartan connection requires only that the <a href="/wiki/Derivative" title="Derivative">derivative</a> of τt be a symmetry of the model space: τ′0 ∈ g, the Lie algebra of G. Typical of Cartan, one motivation for introducing the notion of a Cartan connection was to study the properties of pseudogroups from an infinitesimal point of view. A Cartan connection defines a pseudogroup precisely when the derivative of the transport map τ′ can be <a href="/wiki/Integral" title="Integral">integrated</a>, thus recovering a true (G-valued) transport map between the coordinate systems. There is thus an <a href="/wiki/Integrability_condition" class="mw-redirect" title="Integrability condition">integrability condition</a> at work, and Cartan's method for realizing integrability conditions was to introduce a <a href="/wiki/Differential_form" title="Differential form">differential form</a>. In this case, τ′0 defines a differential form at the point p as follows. For a curve γ(t) = pt in M starting at p, we can associate the <a href="/wiki/Tangent_vector" title="Tangent vector">tangent vector</a> X, as well as a transport map τtγ. Taking the derivative determines a linear map <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\mapsto \left.{\frac {d}{dt}}\tau _{t}^{\gamma }\right|_{t=0}=\theta (X)\in {\mathfrak {g}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">↦</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msubsup> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\mapsto \left.{\frac {d}{dt}}\tau _{t}^{\gamma }\right|_{t=0}=\theta (X)\in {\mathfrak {g}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a00125d1941969bb90d088de47c54a3ff36cf5ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.079ex; height:5.843ex;" alt="{\displaystyle X\mapsto \left.{\frac {d}{dt}}\tau _{t}^{\gamma }\right|_{t=0}=\theta (X)\in {\mathfrak {g}}.}"></dd></dl> So θ defines a g-valued differential 1-form on M. This form, however, is dependent on the choice of parametrized coordinate system. If h : U → H is an H-relation between two parametrized coordinate systems φ and φ′, then the corresponding values of θ are also related by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{p}^{\prime }=Ad(h_{p}^{-1})\theta _{p}+h_{p}^{*}\omega _{H},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>=</mo> <mi>A</mi> <mi>d</mi> <mo stretchy="false">(</mo> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{p}^{\prime }=Ad(h_{p}^{-1})\theta _{p}+h_{p}^{*}\omega _{H},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08aaef39c6b177d209d62462c25ed4019b8461cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.86ex; height:3.343ex;" alt="{\displaystyle \theta _{p}^{\prime }=Ad(h_{p}^{-1})\theta _{p}+h_{p}^{*}\omega _{H},}"></dd></dl> where ωH is the Maurer-Cartan form of H. <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=6" title="Edit section: Formal definition">edit</a>]</div> A Cartan geometry modelled on a homogeneous space G/H can be viewed as a deformation of this geometry which allows for the presence of curvature. For example: <ul><li>a <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian manifold</a> can be seen as a deformation of <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>;</li> <li>a <a href="/wiki/Lorentzian_manifold" class="mw-redirect" title="Lorentzian manifold">Lorentzian manifold</a> can be seen as a deformation of <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>;</li> <li>a <a href="/wiki/Conformal_geometry" title="Conformal geometry">conformal manifold</a> can be seen as a deformation of the <a href="/wiki/Conformal_geometry" title="Conformal geometry">conformal sphere</a>;</li> <li>a manifold equipped with an <a href="/wiki/Affine_connection" title="Affine connection">affine connection</a> can be seen as a deformation of an <a href="/wiki/Affine_space" title="Affine space">affine space</a>.</li></ul> There are two main approaches to the definition. In both approaches, M is a smooth manifold of dimension n, H is a Lie group of dimension m, with Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}">, and G is a Lie group of dimension n+m, with Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}">, containing H as a subgroup. <div class="mw-heading mw-heading3"><h3 id="Definition_via_gauge_transitions">Definition via gauge transitions</h3>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=7" title="Edit section: Definition via gauge transitions">edit</a>]</div> A Cartan connection consists<a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a> of a <a href="/wiki/Atlas_(topology)" title="Atlas (topology)">coordinate atlas</a> of open sets U in M, along with a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}">-valued 1-form θU defined on each chart such that <ol><li>θU : TU → <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}">.</li> <li>θU mod <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}"> : TuU → <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}/{\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}/{\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c01fb82f4ee7669dacfb5105a1a8c57b04324af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.545ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {g}}/{\mathfrak {h}}}"> is a linear isomorphism for every u ∈ U.</li> <li>For any pair of charts U and V in the atlas, there is a smooth mapping h : U ∩ V → H such that</li></ol> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{V}=Ad(h^{-1})\theta _{U}+h^{*}\omega _{H},\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mi>A</mi> <mi>d</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>+</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{V}=Ad(h^{-1})\theta _{U}+h^{*}\omega _{H},\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/672bcae5baec127e70782b8a2e9bb24beabf37be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.113ex; height:3.176ex;" alt="{\displaystyle \theta _{V}=Ad(h^{-1})\theta _{U}+h^{*}\omega _{H},\,}"></dd></dl></dd> <dd>where ωH is the <a href="/wiki/Maurer-Cartan_form" class="mw-redirect" title="Maurer-Cartan form">Maurer-Cartan form</a> of H.</dd></dl> By analogy with the case when the θU came from coordinate systems, condition 3 means that φU is related to φV by h. The curvature of a Cartan connection consists of a system of 2-forms defined on the charts, given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{U}=d\theta _{U}+{\tfrac {1}{2}}[\theta _{U},\theta _{U}].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>=</mo> <mi>d</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">[</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{U}=d\theta _{U}+{\tfrac {1}{2}}[\theta _{U},\theta _{U}].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eef2835afcb9abd96e19cb9a859346f652852cfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.708ex; height:3.509ex;" alt="{\displaystyle \Omega _{U}=d\theta _{U}+{\tfrac {1}{2}}[\theta _{U},\theta _{U}].}"></dd></dl> ΩU satisfy the compatibility condition: <dl><dd>If the forms θU and θV are related by a function h : U ∩ V → H, as above, then ΩV = Ad(h−1) ΩU</dd></dl> The definition can be made independent of the coordinate systems by forming the <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">quotient space</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(\coprod _{U}U\times H)/\sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <munder> <mo>∐</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </munder> <mi>U</mi> <mo>×</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(\coprod _{U}U\times H)/\sim }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d59639e18b9c1f728197b47d9fc3b11b933d35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.312ex; height:5.509ex;" alt="{\displaystyle P=(\coprod _{U}U\times H)/\sim }"></dd></dl> of the disjoint union over all U in the atlas. The <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> ~ is defined on pairs (x,h1) ∈ U1 × H and (x, h2) ∈ U2 × H, by <dl><dd>(x,h1) ~ (x, h2) if and only if x ∈ U1 ∩ U2, θU1 is related to θU2 by h, and h2 = h(x)−1 h1.</dd></dl> Then P is a <a href="/wiki/Principal_bundle" title="Principal bundle">principal H-bundle</a> on M, and the compatibility condition on the connection forms θU implies that they lift to a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}">-valued 1-form η defined on P (see below). <div class="mw-heading mw-heading3"><h3 id="Definition_via_absolute_parallelism">Definition via absolute parallelism</h3>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=8" title="Edit section: Definition via absolute parallelism">edit</a>]</div> Let P be a principal H bundle over M. Then a Cartan connection<a href="#cite_note-9">[9]</a> is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}">-valued 1-form η on P such that <ol><li>for all h in H, Ad(h)Rh*η = η</li> <li>for all ξ in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}">, η(Xξ) = ξ</li> <li>for all p in P, the restriction of η defines a linear isomorphism from the tangent space TpP to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}">.</li></ol> The last condition is sometimes called the Cartan condition: it means that η defines an absolute parallelism on P. The second condition implies that η is already injective on vertical vectors and that the 1-form η mod <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}">, with values in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}/{\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}/{\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c01fb82f4ee7669dacfb5105a1a8c57b04324af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.545ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {g}}/{\mathfrak {h}}}">, is horizontal. The vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}/{\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}/{\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c01fb82f4ee7669dacfb5105a1a8c57b04324af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.545ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {g}}/{\mathfrak {h}}}"> is a <a href="/wiki/Group_representation" title="Group representation">representation</a> of H using the adjoint representation of H on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}">, and the first condition implies that η mod <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}"> is equivariant. Hence it defines a bundle homomorphism from TM to the associated bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\times _{H}{\mathfrak {g}}/{\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\times _{H}{\mathfrak {g}}/{\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3e6be28fbbb6b878f6fae2534a647f81bbbd89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.823ex; height:2.843ex;" alt="{\displaystyle P\times _{H}{\mathfrak {g}}/{\mathfrak {h}}}">. The Cartan condition is equivalent to this bundle homomorphism being an isomorphism, so that η mod <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}"> is a <a href="/wiki/Solder_form" title="Solder form">solder form</a>. The curvature of a Cartan connection is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}">-valued 2-form Ω defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =d\eta +{\tfrac {1}{2}}[\eta \wedge \eta ].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mi>d</mi> <mi>η</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">[</mo> <mi>η</mi> <mo>∧</mo> <mi>η</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =d\eta +{\tfrac {1}{2}}[\eta \wedge \eta ].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32c352ea14e50aab6cec3a0d0b24b64997d8743c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.522ex; height:3.509ex;" alt="{\displaystyle \Omega =d\eta +{\tfrac {1}{2}}[\eta \wedge \eta ].}"></dd></dl> Note that this definition of a Cartan connection looks very similar to that of a <a href="/wiki/Principal_connection" class="mw-redirect" title="Principal connection">principal connection</a>. There are several important differences, however. First, the 1-form η takes values in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}">, but is only equivariant under the action of H. Indeed, it cannot be equivariant under the full group G because there is no G bundle and no G action. Secondly, the 1-form is an absolute parallelism, which intuitively means that η yields information about the behavior of additional directions in the principal bundle (rather than simply being a projection operator onto the vertical space). Concretely, the existence of a solder form binds (or solders) the Cartan connection to the underlying <a href="/wiki/Differential_topology" title="Differential topology">differential topology</a> of the manifold. An intuitive interpretation of the Cartan connection in this form is that it determines a fracturing of the tautological principal bundle associated to a Klein geometry. Thus Cartan geometries are deformed analogues of Klein geometries. This deformation is roughly a prescription for attaching a copy of the model space G/H to each point of M and thinking of that model space as being tangent to (and infinitesimally identical with) the manifold at a point of contact. The fibre of the tautological bundle G → G/H of the Klein geometry at the point of contact is then identified with the fibre of the bundle P. Each such fibre (in G) carries a Maurer-Cartan form for G, and the Cartan connection is a way of assembling these Maurer-Cartan forms gathered from the points of contact into a coherent 1-form η defined on the whole bundle. The fact that only elements of H contribute to the Maurer-Cartan equation Ad(h)Rh*η = η has the intuitive interpretation that any other elements of G would move the model space away from the point of contact, and so no longer be tangent to the manifold. From the Cartan connection, defined in these terms, one can recover a Cartan connection as a system of 1-forms on the manifold (as in the gauge definition) by taking a collection of <a href="/wiki/Locally_trivial" class="mw-redirect" title="Locally trivial">local trivializations</a> of P given as sections sU : U → P and letting θU = s*η be the <a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">pullbacks</a> of the Cartan connection along the sections. <div class="mw-heading mw-heading3"><h3 id="As_principal_connections">As principal connections</h3>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=9" title="Edit section: As principal connections">edit</a>]</div> Another way in which to define a Cartan connection is as a <a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">principal connection</a> on a certain principal G-bundle. From this perspective, a Cartan connection consists of <ul><li>a principal G-bundle Q over M</li> <li>a principal G-connection α on Q (the Cartan connection)</li> <li>a principal H-subbundle P of Q (i.e., a reduction of structure group)</li></ul> such that the <a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">pullback</a> η of α to P satisfies the Cartan condition. The principal connection α on Q can be recovered from the form η by taking Q to be the associated bundle P ×H G. Conversely, the form η can be recovered from α by pulling back along the inclusion P ⊂ Q. Since α is a principal connection, it induces a <a href="/wiki/Ehresmann_connection" title="Ehresmann connection">connection</a> on any <a href="/wiki/Associated_bundle" title="Associated bundle">associated bundle</a> to Q. In particular, the bundle Q ×G G/H of homogeneous spaces over M, whose fibers are copies of the model space G/H, has a connection. The reduction of structure group to H is equivalently given by a section s of E = Q ×G G/H. The fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\times _{H}{\mathfrak {g}}/{\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\times _{H}{\mathfrak {g}}/{\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3e6be28fbbb6b878f6fae2534a647f81bbbd89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.823ex; height:2.843ex;" alt="{\displaystyle P\times _{H}{\mathfrak {g}}/{\mathfrak {h}}}"> over x in M may be viewed as the tangent space at s(x) to the fiber of Q ×G G/H over x. Hence the Cartan condition has the intuitive interpretation that the model spaces are tangent to M along the section s. Since this identification of tangent spaces is induced by the connection, the marked points given by s always move under parallel transport. <div class="mw-heading mw-heading3"><h3 id="Definition_by_an_Ehresmann_connection">Definition by an Ehresmann connection</h3>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=10" title="Edit section: Definition by an Ehresmann connection">edit</a>]</div> Yet another way to define a Cartan connection is with an <a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann connection</a> on the bundle E = Q ×G G/H of the preceding section.<a href="#cite_note-10">[10]</a> A Cartan connection then consists of <ul><li>A <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundle</a> π : E → M with fibre G/H and vertical space VE ⊂ TE.</li> <li>A section s : M → E.</li> <li>A <a href="/wiki/Ehresmann_connection#Associated_bundles" title="Ehresmann connection">G-connection</a> θ : TE → VE such that</li></ul> <dl><dd><dl><dd>s*θx : TxM → Vs(x)E is a linear isomorphism of vector spaces for all x ∈ M.</dd></dl></dd></dl> This definition makes rigorous the intuitive ideas presented in the introduction. First, the preferred section s can be thought of as identifying a point of contact between the manifold and the tangent space. The last condition, in particular, means that the tangent space of M at x is isomorphic to the tangent space of the model space at the point of contact. So the model spaces are, in this way, tangent to the manifold. <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Model_space_development.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Model_space_development.svg/220px-Model_space_development.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Model_space_development.svg/330px-Model_space_development.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/09/Model_space_development.svg/440px-Model_space_development.svg.png 2x" data-file-width="625" data-file-height="625" /></a><figcaption>Development of a curve into the model space at x0</figcaption></figure> This definition also brings prominently into focus the idea of <a href="/wiki/Development_(differential_geometry)" title="Development (differential geometry)">development</a>. If xt is a curve in M, then the Ehresmann connection on E supplies an associated <a href="/wiki/Parallel_transport" title="Parallel transport">parallel transport</a> map τt : Ext → Ex0 from the fibre over the endpoint of the curve to the fibre over the initial point. In particular, since E is equipped with a preferred section s, the points s(xt) transport back to the fibre over x0 and trace out a curve in Ex0. This curve is then called the development of the curve xt. To show that this definition is equivalent to the others above, one must introduce a suitable notion of a <a href="/wiki/Moving_frame" title="Moving frame">moving frame</a> for the bundle E. In general, this is possible for any G-connection on a fibre bundle with structure group G. See <a href="/wiki/Ehresmann_connection#Associated_bundles" title="Ehresmann connection">Ehresmann connection#Associated bundles</a> for more details. <div class="mw-heading mw-heading2"><h2 id="Special_Cartan_connections">Special Cartan connections</h2>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=11" title="Edit section: Special Cartan connections">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Reductive_Cartan_connections">Reductive Cartan connections</h3>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=12" title="Edit section: Reductive Cartan connections">edit</a>]</div> Let P be a principal H-bundle on M, equipped with a Cartan connection η : TP → <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}"> is a <a href="/wiki/Reductive_Lie_algebra" title="Reductive Lie algebra">reductive module</a> for H, meaning that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}"> admits an <a href="/wiki/Adjoint_representation" title="Adjoint representation">Ad</a>(H)-invariant splitting of vector spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus {\mathfrak {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus {\mathfrak {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b09ced9cceed92c4a587460c82fdb880400ddbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.104ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus {\mathfrak {m}}}">, then the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0e9162e96758157a34a6e44967288b481a7cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}">-component of η generalizes the solder form for an <a href="/wiki/Affine_connection" title="Affine connection">affine connection</a>.<a href="#cite_note-11">[11]</a> In detail, η splits into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0e9162e96758157a34a6e44967288b481a7cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}"> components: <dl><dd>η = η<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}"> + η<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0e9162e96758157a34a6e44967288b481a7cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}">.</dd></dl> Note that the 1-form η<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}"> is a principal H-connection on the original Cartan bundle P. Moreover, the 1-form η<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0e9162e96758157a34a6e44967288b481a7cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}"> satisfies: <dl><dd>η<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0e9162e96758157a34a6e44967288b481a7cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}">(X) = 0 for every vertical vector X ∈ TP. (η<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0e9162e96758157a34a6e44967288b481a7cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}"> is horizontal.)</dd> <dd>Rh*η<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0e9162e96758157a34a6e44967288b481a7cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}"> = <a href="/wiki/Adjoint_representation" title="Adjoint representation">Ad</a>(h−1)η<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0e9162e96758157a34a6e44967288b481a7cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}"> for every h ∈ H. (η<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0e9162e96758157a34a6e44967288b481a7cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}"> is equivariant under the right H-action.)</dd></dl> In other words, η is a <a href="/wiki/Solder_form" title="Solder form">solder form</a> for the bundle P. Hence, P equipped with the form η<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0e9162e96758157a34a6e44967288b481a7cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}"> defines a (first order) <a href="/wiki/G-structure" class="mw-redirect" title="G-structure">H-structure</a> on M. The form η<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}"> defines a connection on the H-structure. <div class="mw-heading mw-heading3"><h3 id="Parabolic_Cartan_connections">Parabolic Cartan connections</h3>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=13" title="Edit section: Parabolic Cartan connections">edit</a>]</div> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}"> is a <a href="/wiki/Semisimple_Lie_algebra" title="Semisimple Lie algebra">semisimple Lie algebra</a> with <a href="/wiki/Parabolic_Lie_algebra" title="Parabolic Lie algebra">parabolic subalgebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}"> (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}"> contains a <a href="/wiki/Borel_subalgebra" title="Borel subalgebra">maximal solvable subalgebra</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}">) and G and P are associated Lie groups, then a Cartan connection modelled on (G,P,<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}">,<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}">) is called a parabolic Cartan geometry, or simply a parabolic geometry. A distinguishing feature of parabolic geometries is a Lie algebra structure on its <a href="/wiki/Cotangent_space" title="Cotangent space">cotangent spaces</a>: this arises because the perpendicular subspace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}">⊥ of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}"> with respect to the <a href="/wiki/Killing_form" title="Killing form">Killing form</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}"> is a subalgebra of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}">, and the Killing form induces a natural duality between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}">⊥ and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}/{\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}/{\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c53703d001ad8e3fa4150a0610a13633dd080e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.497ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {g}}/{\mathfrak {p}}}">. Thus the bundle associated to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}">⊥ is isomorphic to the <a href="/wiki/Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a>. Parabolic geometries include many of those of interest in research and applications of Cartan connections, such as the following examples: <ul><li><a href="/wiki/Conformal_connection" title="Conformal connection">Conformal connections</a>: Here G = SO(p+1,q+1), and P is the stabilizer of a null ray in Rn+2.</li> <li><a href="/wiki/Projective_connection" title="Projective connection">Projective connections</a>: Here G = PGL(n+1) and P is the stabilizer of a point in RPn.</li> <li><a href="/wiki/CR_structure" class="mw-redirect" title="CR structure">CR structures</a> and Cartan-Chern-Tanaka connections: G = PSU(p+1,q+1), P = stabilizer of a point on the projective null <a href="/wiki/Quadric" title="Quadric">hyperquadric</a>.</li> <li>Contact projective connections:<a href="#cite_note-12">[12]</a> Here G = SP(2n+2) and P is the stabilizer of the ray generated by the first standard basis vector in Rn+2.</li> <li>Generic rank 2 distributions on 5-manifolds: Here G = Aut(Os) is the automorphism group of the algebra Os of <a href="/w/index.php?title=Split_octonion&action=edit&redlink=1" class="new" title="Split octonion (page does not exist)">split octonions</a>, a <a href="/wiki/Closed_subgroup" class="mw-redirect" title="Closed subgroup">closed subgroup</a> of SO(3,4), and P is the intersection of G with the stabilizer of the isotropic line spanned by the first standard basis vector in R7 viewed as the purely imaginary split octonions (orthogonal complement of the unit element in Os).<a href="#cite_note-13">[13]</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Associated_differential_operators">Associated differential operators</h2>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=14" title="Edit section: Associated differential operators">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Covariant_differentiation">Covariant differentiation</h3>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=15" title="Edit section: Covariant differentiation">edit</a>]</div> Suppose that M is a Cartan geometry modelled on G/H, and let (Q,α) be the principal G-bundle with connection, and (P,η) the corresponding reduction to H with η equal to the pullback of α. Let V a <a href="/wiki/Group_representation" title="Group representation">representation</a> of G, and form the vector bundle V = Q ×G V over M. Then the principal G-connection α on Q induces a <a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">covariant derivative</a> on V, which is a first order <a href="/wiki/Linear_differential_operator" class="mw-redirect" title="Linear differential operator">linear differential operator</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \colon \Omega _{M}^{0}(\mathbf {V} )\to \Omega _{M}^{1}(\mathbf {V} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>:</mo> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \colon \Omega _{M}^{0}(\mathbf {V} )\to \Omega _{M}^{1}(\mathbf {V} ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6086a78de62c747e0322e6abbd0789b2050b79f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.163ex; height:3.176ex;" alt="{\displaystyle \nabla \colon \Omega _{M}^{0}(\mathbf {V} )\to \Omega _{M}^{1}(\mathbf {V} ),}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{M}^{k}(\mathbf {V} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{M}^{k}(\mathbf {V} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbc757f0d8f136fe0b0d0d4f2154b748bb15082" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.466ex; height:3.176ex;" alt="{\displaystyle \Omega _{M}^{k}(\mathbf {V} )}"> denotes the space of <a href="/wiki/Vector_valued_differential_form" class="mw-redirect" title="Vector valued differential form">k-forms on M with values in V</a> so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{M}^{0}(\mathbf {V} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{M}^{0}(\mathbf {V} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d617831630258eedb12bf3bf2914ab455a0b6ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.466ex; height:3.176ex;" alt="{\displaystyle \Omega _{M}^{0}(\mathbf {V} )}"> is the space of sections of V and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{M}^{1}(\mathbf {V} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{M}^{1}(\mathbf {V} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b61f346097cd438e4498c7a50eb536cfdc8a2a10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.466ex; height:3.176ex;" alt="{\displaystyle \Omega _{M}^{1}(\mathbf {V} )}"> is the space of sections of Hom(TM,V). For any section v of V, the contraction of the covariant derivative ∇v with a vector field X on M is denoted ∇Xv and satisfies the following Leibniz rule: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{X}(fv)=df(X)v+f\nabla _{X}v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo>+</mo> <mi>f</mi> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{X}(fv)=df(X)v+f\nabla _{X}v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab1314bc8b7f6f4ec8c83d07d54d8e752a91951" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.108ex; height:2.843ex;" alt="{\displaystyle \nabla _{X}(fv)=df(X)v+f\nabla _{X}v}"></dd></dl> for any smooth function f on M. The covariant derivative can also be constructed from the Cartan connection η on P. In fact, constructing it in this way is slightly more general in that V need not be a fully fledged representation of G.<a href="#cite_note-14">[14]</a> Suppose instead that V is a (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}">, H)-module: a representation of the group H with a compatible representation of the Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}">. Recall that a section v of the induced vector bundle V over M can be thought of as an H-equivariant map P → V. This is the point of view we shall adopt. Let X be a vector field on M. Choose any right-invariant lift <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b968141b314f4de17f5e63f18dcdc126352bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.509ex;" alt="{\displaystyle {\bar {X}}}"> to the tangent bundle of P. Define <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{X}v=dv({\bar {X}})+\eta ({\bar {X}})\cdot v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mi>v</mi> <mo>=</mo> <mi>d</mi> <mi>v</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{X}v=dv({\bar {X}})+\eta ({\bar {X}})\cdot v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a233d867f4faca2f640625d0017fdddd9ae864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.533ex; height:3.009ex;" alt="{\displaystyle \nabla _{X}v=dv({\bar {X}})+\eta ({\bar {X}})\cdot v}">.</dd></dl> In order to show that ∇v is well defined, it must: <ol><li>be independent of the chosen lift <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b968141b314f4de17f5e63f18dcdc126352bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.509ex;" alt="{\displaystyle {\bar {X}}}"></li> <li>be equivariant, so that it descends to a section of the bundle V.</li></ol> For (1), the ambiguity in selecting a right-invariant lift of X is a transformation of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\mapsto X+X_{\xi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">↦</mo> <mi>X</mi> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\mapsto X+X_{\xi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18236522c187c16562ca4c814262c894c1432778" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.299ex; height:2.843ex;" alt="{\displaystyle X\mapsto X+X_{\xi }}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{\xi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{\xi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a52c9222d02ee54b7699ba0be2bdeb9424b4afee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.885ex; height:2.843ex;" alt="{\displaystyle X_{\xi }}"> is the right-invariant vertical vector field induced from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi \in {\mathfrak {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36cd53281a600b622ce02d2e1c2c78ac2fe59cad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.082ex; height:2.509ex;" alt="{\displaystyle \xi \in {\mathfrak {h}}}">. So, calculating the covariant derivative in terms of the new lift <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}+X_{\xi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}+X_{\xi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb88120a2c457e788c74effba386825c33d967e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.705ex; height:3.176ex;" alt="{\displaystyle {\bar {X}}+X_{\xi }}">, one has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{X}v=dv({\bar {X}}+X_{\xi })+\eta ({\bar {X}}+X_{\xi }))\cdot v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mi>v</mi> <mo>=</mo> <mi>d</mi> <mi>v</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{X}v=dv({\bar {X}}+X_{\xi })+\eta ({\bar {X}}+X_{\xi }))\cdot v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29f03b68272d55f1c62af7ae560d15a52a97963b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.888ex; height:3.176ex;" alt="{\displaystyle \nabla _{X}v=dv({\bar {X}}+X_{\xi })+\eta ({\bar {X}}+X_{\xi }))\cdot v}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =dv({\bar {X}})+dv(X_{\xi })+\eta ({\bar {X}})\cdot v+\xi \cdot v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>d</mi> <mi>v</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>v</mi> <mo>+</mo> <mi>ξ</mi> <mo>⋅</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =dv({\bar {X}})+dv(X_{\xi })+\eta ({\bar {X}})\cdot v+\xi \cdot v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc3c6d1eea1d46f61609b5bb1c2493ee86613a16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.747ex; height:3.176ex;" alt="{\displaystyle =dv({\bar {X}})+dv(X_{\xi })+\eta ({\bar {X}})\cdot v+\xi \cdot v}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =dv({\bar {X}})+\eta ({\bar {X}})\cdot v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>d</mi> <mi>v</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =dv({\bar {X}})+\eta ({\bar {X}})\cdot v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/804661e5eb6b9476555182ef6af39d64843ca877" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.192ex; height:3.009ex;" alt="{\displaystyle =dv({\bar {X}})+\eta ({\bar {X}})\cdot v}"></dd></dl> since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \cdot v+dv(X_{\xi })=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>⋅</mo> <mi>v</mi> <mo>+</mo> <mi>d</mi> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi \cdot v+dv(X_{\xi })=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62bef43ee1d87a4a0d26d8cfa7a4e53583b28816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.976ex; height:3.009ex;" alt="{\displaystyle \xi \cdot v+dv(X_{\xi })=0}"> by taking the differential of the equivariance property <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\cdot R_{h}^{*}v=v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>⋅</mo> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mi>v</mi> <mo>=</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\cdot R_{h}^{*}v=v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daf171cd00ff71b0949c39902ea0ac64872f0b5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.315ex; height:2.843ex;" alt="{\displaystyle h\cdot R_{h}^{*}v=v}"> at h equal to the identity element. For (2), observe that since v is equivariant and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b968141b314f4de17f5e63f18dcdc126352bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.509ex;" alt="{\displaystyle {\bar {X}}}"> is right-invariant, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dv({\bar {X}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>v</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dv({\bar {X}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/838a63e4631666f274f922e28443cb2600fd19d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.133ex; height:3.009ex;" alt="{\displaystyle dv({\bar {X}})}"> is equivariant. On the other hand, since η is also equivariant, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta ({\bar {X}})\cdot v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta ({\bar {X}})\cdot v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f88ab78e9bef655b9bbdb98ab93761c5d1182ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.765ex; height:3.009ex;" alt="{\displaystyle \eta ({\bar {X}})\cdot v}"> is equivariant as well. <div class="mw-heading mw-heading3"><h3 id="The_fundamental_or_universal_derivative">The fundamental or universal derivative</h3>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=16" title="Edit section: The fundamental or universal derivative">edit</a>]</div> Suppose that V is only a representation of the subgroup H and not necessarily the larger group G. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ^{k}(P,V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ^{k}(P,V)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e7cc4d19f0950b7c203a021353a513c302e8a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.143ex; height:3.176ex;" alt="{\displaystyle \Omega ^{k}(P,V)}"> be the space of V-valued differential k-forms on P. In the presence of a Cartan connection, there is a canonical isomorphism <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon \Omega ^{k}(P,V)\cong \Omega ^{0}(P,V\otimes \bigwedge \nolimits ^{k}{\mathfrak {g}}^{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>≅</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>V</mi> <mo>⊗</mo> <msup> <mo movablelimits="false">⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \colon \Omega ^{k}(P,V)\cong \Omega ^{0}(P,V\otimes \bigwedge \nolimits ^{k}{\mathfrak {g}}^{*})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d201cfd52e06ae7ce33dfedc2e1f42c6b54f9213" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:33.027ex; height:4.343ex;" alt="{\displaystyle \varphi \colon \Omega ^{k}(P,V)\cong \Omega ^{0}(P,V\otimes \bigwedge \nolimits ^{k}{\mathfrak {g}}^{*})}"></dd></dl> given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (\beta )(\xi _{1},\xi _{2},\dots ,\xi _{k})=\beta (\eta ^{-1}(\xi _{1}),\dots ,\eta ^{-1}(\xi _{k}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>β</mi> <mo stretchy="false">(</mo> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (\beta )(\xi _{1},\xi _{2},\dots ,\xi _{k})=\beta (\eta ^{-1}(\xi _{1}),\dots ,\eta ^{-1}(\xi _{k}))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b73c73b7bd329cba514c9a50ba00b74942fa1977" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.165ex; height:3.176ex;" alt="{\displaystyle \varphi (\beta )(\xi _{1},\xi _{2},\dots ,\xi _{k})=\beta (\eta ^{-1}(\xi _{1}),\dots ,\eta ^{-1}(\xi _{k}))}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \in \Omega ^{k}(P,V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>∈</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \in \Omega ^{k}(P,V)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb668a6a23eb681f4aa10a2c7c125efe2981312" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.315ex; height:3.176ex;" alt="{\displaystyle \beta \in \Omega ^{k}(P,V)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{j}\in {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{j}\in {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7701ca61b61dda7d8ec785ffa323c15b808b6d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.941ex; height:2.843ex;" alt="{\displaystyle \xi _{j}\in {\mathfrak {g}}}">. For each k, the exterior derivative is a first order operator differential operator <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\colon \Omega ^{k}(P,V)\rightarrow \Omega ^{k+1}(P,V)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>:</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\colon \Omega ^{k}(P,V)\rightarrow \Omega ^{k+1}(P,V)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d44a1603fe631dda457d6df38568b011635528ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.637ex; height:3.176ex;" alt="{\displaystyle d\colon \Omega ^{k}(P,V)\rightarrow \Omega ^{k+1}(P,V)\,}"></dd></dl> and so, for k=0, it defines a differential operator <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \circ d\colon \Omega ^{0}(P,V)\rightarrow \Omega ^{0}(P,V\otimes {\mathfrak {g}}^{*}).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>∘</mo> <mi>d</mi> <mo>:</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>V</mi> <mo>⊗</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \circ d\colon \Omega ^{0}(P,V)\rightarrow \Omega ^{0}(P,V\otimes {\mathfrak {g}}^{*}).\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cca00e353918d5747836f8ac3bdcc2d3a7bbb54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.895ex; height:3.176ex;" alt="{\displaystyle \varphi \circ d\colon \Omega ^{0}(P,V)\rightarrow \Omega ^{0}(P,V\otimes {\mathfrak {g}}^{*}).\,}"></dd></dl> Because η is equivariant, if v is equivariant, so is Dv := φ(dv). It follows that this composite descends to a first order differential operator D from sections of V=P×HV to sections of the bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\times _{H}(\mathbf {V} \otimes {\mathfrak {g}}^{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>⊗</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\times _{H}(\mathbf {V} \otimes {\mathfrak {g}}^{*})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efef7640f3eaf6c228442397b6f91a0a654c11d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.172ex; height:2.843ex;" alt="{\displaystyle P\times _{H}(\mathbf {V} \otimes {\mathfrak {g}}^{*})}">. This is called the fundamental or universal derivative, or fundamental D-operator. <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=17" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> Although Cartan only began formalizing this theory in particular cases in the 1920s (<a href="#CITEREFCartan1926">Cartan 1926</a>), he made much use of the general idea much earlier. The high point of his remarkable 1910 paper on <a href="/wiki/Pfaffian_system" class="mw-redirect" title="Pfaffian system">Pfaffian systems</a> in five variables is the construction of a Cartan connection modelled on a 5-dimensional homogeneous space for the <a href="/wiki/Exceptional_Lie_group" class="mw-redirect" title="Exceptional Lie group">exceptional Lie group</a> <a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G2</a>, which he and Engels had discovered independently in 1894. </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFChevalley1946">Chevalley 1946</a>, p. 110. </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> See R. Hermann (1983), Appendix 1–3 to <a href="#CITEREFCartan1951">Cartan (1951)</a>. </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> This appears to be Cartan's way of viewing the connection. Cf. <a href="#CITEREFCartan1923">Cartan 1923</a>, p. 362; <a href="#CITEREFCartan1924">Cartan 1924</a>, p. 208 especially ..un repère définissant un système de coordonnées projectives...; <a href="#CITEREFCartan1951">Cartan 1951</a>, p. 34. Modern readers can arrive at various interpretations of these statements, cf. Hermann's 1983 notes in <a href="#CITEREFCartan1951">Cartan 1951</a>, pp. 384–385, 477. </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> More precisely, hp is required to be in the <a href="/wiki/Isotropy_group" class="mw-redirect" title="Isotropy group">isotropy group</a> of φp(p), which is a group in G isomorphic to H. </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> In general, this is not the rolling map described in the motivation, although it is related. </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFSharpe1997">Sharpe 1997</a>. </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFLumiste2001a">Lumiste 2001a</a>. </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> This is the standard definition. Cf. Hermann (1983), Appendix 2 to <a href="#CITEREFCartan1951">Cartan 1951</a>; <a href="#CITEREFKobayashi1970">Kobayashi 1970</a>, p. 127; <a href="#CITEREFSharpe1997">Sharpe 1997</a>; <a href="#CITEREFSlovák1997">Slovák 1997</a>. </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="#CITEREFEhresmann1950">Ehresmann 1950</a>, <a href="#CITEREFKobayashi1957">Kobayashi 1957</a>, <a href="#CITEREFLumiste2001b">Lumiste 2001b</a>. </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> For a treatment of affine connections from this point of view, see <a href="#CITEREFKobayashiNomizu1996">Kobayashi & Nomizu (1996</a>, Volume 1). </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> See, for example, <a href="#CITEREFFox2005">Fox (2005)</a>. </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <a href="#CITEREFSagerschnig2006">Sagerschnig 2006</a>; <a href="#CITEREFČapSagerschnig2009">Čap & Sagerschnig 2009</a>. </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> See, for instance, <a href="#CITEREFČapGover2002">Čap & Gover (2002</a>, Definition 2.4). </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=18" title="Edit section: References">edit</a>]</div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFČapGover2002" class="citation cs2">Čap, Andreas; Gover, A. Rod (2002), "Tractor calculi for parabolic geometries]", Transactions of the American Mathematical Society, 354 (4): 1511–1548, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9947-01-02909-9">10.1090/S0002-9947-01-02909-9</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Tractor+calculi+for+parabolic+geometries%5D&rft.volume=354&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1511-%3C%2Fspan%3E1548&rft.date=2002&rft_id=info%3Adoi%2F10.1090%2FS0002-9947-01-02909-9&rft.aulast=%C4%8Cap&rft.aufirst=Andreas&rft.au=Gover%2C+A.+Rod&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartan+connection" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFČapSagerschnig2009" class="citation cs2">Čap, A.; Sagerschnig, K. 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+Scientifiques+de+l%27%C3%89cole+Normale+Sup%C3%A9rieure&rft.atitle=Sur+les+vari%C3%A9t%C3%A9s+%C3%A0+connexion+affine+et+la+th%C3%A9orie+de+la+relativit%C3%A9+g%C3%A9n%C3%A9ralis%C3%A9e+%28premi%C3%A8re+partie%29&rft.volume=40&rft.pages=%3Cspan+class%3D%22nowrap%22%3E325-%3C%2Fspan%3E412&rft.date=1923&rft_id=info%3Adoi%2F10.24033%2Fasens.751&rft.aulast=Cartan&rft.aufirst=%C3%89lie&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartan+connection" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCartan1924" class="citation cs2"><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan, Élie</a> (1924), "Sur les variétés à connexion projective", Bulletin de la Société Mathématique de France, 52: 205–241, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Connections+on+a+manifold&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=Kluwer+Academic+Publishers&rft.date=2001&rft.isbn=978-1-55608-010-4&rft.aulast=Lumiste&rft.aufirst=%C3%9C.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DConnections_on_a_manifold&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartan+connection" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSagerschnig2006" class="citation cs2">Sagerschnig, K. (2006), <a rel="nofollow" class="external text" href="http://www.emis.de/journals/AM/06-S/sager.ps.gz">"Split octonions and generic rank two distributions in dimension five"</a>, Archivum Mathematicum, 42 (Suppl): 329–339</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archivum+Mathematicum&rft.atitle=Split+octonions+and+generic+rank+two+distributions+in+dimension+five&rft.volume=42&rft.issue=Suppl&rft.pages=%3Cspan+class%3D%22nowrap%22%3E329-%3C%2Fspan%3E339&rft.date=2006&rft.aulast=Sagerschnig&rft.aufirst=K.&rft_id=http%3A%2F%2Fwww.emis.de%2Fjournals%2FAM%2F06-S%2Fsager.ps.gz&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartan+connection" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSharpe1997" class="citation cs2">Sharpe, R.W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94732-9" title="Special:BookSources/0-387-94732-9"><bdi>0-387-94732-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Geometry%3A+Cartan%27s+Generalization+of+Klein%27s+Erlangen+Program&rft.pub=Springer-Verlag%2C+New+York&rft.date=1997&rft.isbn=0-387-94732-9&rft.aulast=Sharpe&rft.aufirst=R.W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartan+connection" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSlovák1997" class="citation cs2">Slovák, Jan (1997), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220330020554/https://dml.cz/bitstream/handle/10338.dmlcz/108039/ArchMathRetro_042-2006-5_20.pdf">Parabolic Geometries</a> (PDF), Research Lecture Notes, Part of DrSc-dissertation, Masaryk University, archived from <a rel="nofollow" class="external text" href="https://dml.cz/bitstream/handle/10338.dmlcz/108039/ArchMathRetro_042-2006-5_20.pdf">the original</a> (PDF) on March 30, 2022</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Parabolic+Geometries&rft.series=Research+Lecture+Notes&rft.pub=Part+of+DrSc-dissertation%2C+Masaryk+University&rft.date=1997&rft.aulast=Slov%C3%A1k&rft.aufirst=Jan&rft_id=https%3A%2F%2Fdml.cz%2Fbitstream%2Fhandle%2F10338.dmlcz%2F108039%2FArchMathRetro_042-2006-5_20.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartan+connection" class="Z3988">.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Books">Books</h2>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=19" title="Edit section: Books">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKobayashi1972" class="citation cs2">Kobayashi, Shoshichi (1972), Transformations Groups in Differential Geometry (Classics in Mathematics 1995 ed.), Springer-Verlag, Berlin, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-58659-3" title="Special:BookSources/978-3-540-58659-3"><bdi>978-3-540-58659-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Transformations+Groups+in+Differential+Geometry&rft.edition=Classics+in+Mathematics+1995&rft.pub=Springer-Verlag%2C+Berlin&rft.date=1972&rft.isbn=978-3-540-58659-3&rft.aulast=Kobayashi&rft.aufirst=Shoshichi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartan+connection" class="Z3988">.</li></ul> <dl><dd><dl><dd>The section 3. Cartan Connections [pages 127–130] treats conformal and projective connections in a unified manner.</dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Cartan_connection&action=edit&section=20" title="Edit section: External links">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFÜ._Lumiste2001" class="citation cs2"><a href="/wiki/%C3%9Clo_Lumiste" title="Ülo Lumiste">Ü. Lumiste</a> (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Affine_connection">"Affine connection"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Affine+connection&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.au=%C3%9C.+Lumiste&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DAffine_connection&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartan+connection" class="Z3988"></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist 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navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Manifolds" title="Template:Manifolds"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Manifolds" title="Template talk:Manifolds"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Manifolds" title="Special:EditPage/Template:Manifolds"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Manifolds_(Glossary,_List,_Category)273" style="font-size:114%;margin:0 4em"><a href="/wiki/Manifold" title="Manifold">Manifolds</a> (<a href="/wiki/Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>, <a href="/wiki/List_of_manifolds" title="List of manifolds">List</a>, <a href="/wiki/Category:Manifolds" title="Category:Manifolds">Category</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Topological_manifold" title="Topological manifold">Topological manifold</a> <ul><li><a href="/wiki/Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li> <li><a href="/wiki/Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a> <ul><li><a href="/wiki/Differential_structure" title="Differential structure">Differential structure</a></li> <li><a href="/wiki/Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li> <li><a href="/wiki/Submanifold" title="Submanifold">Submanifold</a></li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li> <li><a href="/wiki/Smoothness" title="Smoothness">Smooth map</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results <a href="/wiki/Category:Theorems_in_differential_geometry" title="Category:Theorems in differential geometry">(list)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li> <li><a href="/wiki/Darboux%27s_theorem" title="Darboux's theorem">Darboux's</a></li> <li><a href="/wiki/De_Rham_cohomology#De_Rham's_theorem" title="De Rham cohomology">De Rham's</a></li> <li><a href="/wiki/Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li> <li><a href="/wiki/Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li> <li><a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's</a></li> <li><a href="/wiki/Sard%27s_theorem" title="Sard's theorem">Sard's</a></li> <li><a href="/wiki/Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Smoothness" title="Smoothness">Maps</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differentiable_curve" title="Differentiable curve">Curve</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a> <ul><li><a href="/wiki/Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a> <ul><li><a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li> <li><a href="/wiki/Foliation" title="Foliation">Foliation</a></li> <li><a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li> <li><a href="/wiki/Integral_curve" title="Integral curve">Integral curve</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of manifolds</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Closed_manifold" title="Closed manifold">Closed</a></li> <li><a href="/wiki/Collapsing_manifold" title="Collapsing manifold">Collapsing</a></li> <li><a href="/wiki/Complete_manifold" title="Complete manifold">Complete</a></li> <li>(<a href="/wiki/Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a href="/wiki/Complex_manifold" title="Complex manifold">Complex</a></li> <li>(<a href="/wiki/Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="/wiki/Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li> <li><a href="/wiki/Fibered_manifold" title="Fibered manifold">Fibered</a></li> <li><a href="/wiki/Finsler_manifold" title="Finsler manifold">Finsler</a></li> <li>(<a href="/wiki/Almost_flat_manifold" title="Almost flat manifold">Almost</a>) <a href="/wiki/Flat_manifold" title="Flat manifold">Flat</a></li> <li><a href="/wiki/G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li> <li><a href="/wiki/Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li> <li><a href="/wiki/Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li> <li><a href="/wiki/Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li> <li><a href="/wiki/Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a> <ul><li><a href="/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li> <li><a href="/wiki/Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li> <li><a href="/wiki/Nilmanifold" title="Nilmanifold">Nilmanifold</a></li> <li><a href="/wiki/Orientability" title="Orientability">Oriented</a></li> <li><a href="/wiki/Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li> <li><a href="/wiki/Poisson_manifold" title="Poisson manifold">Poisson</a></li> <li><a href="/wiki/Prime_manifold" title="Prime manifold">Prime</a></li> <li><a href="/wiki/Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li> <li><a href="/wiki/Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li> <li>(<a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo−</a>, <a href="/wiki/Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub−</a>) <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li> <li><a href="/wiki/Rizza_manifold" title="Rizza manifold">Rizza</a></li> <li>(<a href="/wiki/Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li> <li><a href="/wiki/Tame_manifold" title="Tame manifold">Tame</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Tensor" title="Tensor">Tensors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Vectors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li> <li><a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a> <ul><li><a href="/wiki/Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li> <li><a href="/wiki/Vector_flow" title="Vector flow">Vector flow</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li> <li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Cotangent_space" title="Cotangent space">Cotangent space</a> <ul><li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a> <ul><li><a href="/wiki/Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Interior_product" title="Interior product">Interior product</a></li> <li><a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a> <ul><li><a href="/wiki/Ricci_flow" title="Ricci flow">flow</a></li></ul></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a> <ul><li><a href="/wiki/Tensor_density" title="Tensor density">density</a></li></ul></li> <li><a href="/wiki/Volume_form" title="Volume form">Volume form</a></li> <li><a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fiber_bundle" title="Fiber bundle">Bundles</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li> <li><a href="/wiki/Affine_bundle" title="Affine bundle">Affine</a></li> <li><a href="/wiki/Associated_bundle" title="Associated bundle">Associated</a></li> <li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li> <li><a href="/wiki/Dual_bundle" title="Dual bundle">Dual</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber</a></li> <li>(<a href="/wiki/Cofibration" title="Cofibration">Co</a>) <a href="/wiki/Fibration" title="Fibration">Fibration</a></li> <li><a href="/wiki/Jet_bundle" title="Jet bundle">Jet</a></li> <li><a href="/wiki/Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li> <li>(<a href="/wiki/Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="/wiki/Normal_bundle" title="Normal bundle">Normal</a></li> <li><a href="/wiki/Principal_bundle" title="Principal bundle">Principal</a></li> <li><a href="/wiki/Spinor_bundle" title="Spinor bundle">Spinor</a></li> <li><a href="/wiki/Subbundle" title="Subbundle">Subbundle</a></li> <li><a href="/wiki/Tangent_bundle" title="Tangent bundle">Tangent</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor</a></li> <li><a href="/wiki/Vector_bundle" title="Vector bundle">Vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">Connections</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine</a></li> <li><a class="mw-selflink selflink">Cartan</a></li> <li><a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Form</a></li> <li><a href="/wiki/Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li> <li><a href="/wiki/Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li> <li><a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li> <li><a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li> <li><a href="/wiki/Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li> <li><a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a></li> <li><a href="/wiki/Moving_frame" title="Moving frame">Moving frame</a></li> <li><a href="/wiki/Singularity_theory" title="Singularity theory">Singularity theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifold</a></li> <li><a href="/wiki/Diffeology" title="Diffeology">Diffeology</a></li> <li><a href="/wiki/Diffiety" title="Diffiety">Diffiety</a></li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a> <ul><li><a href="/wiki/Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li> <li><a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li> <li><a href="/wiki/Stratifold" title="Stratifold">Stratifold</a></li> <li><a href="/wiki/Supermanifold" title="Supermanifold">Supermanifold</a></li> <li><a href="/wiki/Stratified_space" title="Stratified space">Stratified space</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Cartan_connection&oldid=1236104926">https://en.wikipedia.org/w/index.php?title=Cartan_connection&oldid=1236104926</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Connection_(mathematics)" title="Category:Connection 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