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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Soldering_of_a_fibre_bundle" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Soldering_of_a_fibre_bundle"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Soldering of a fibre bundle</span> </div> </a> <ul id="toc-Soldering_of_a_fibre_bundle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_cases" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Special_cases"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Special cases</span> </div> </a> <button aria-controls="toc-Special_cases-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Special cases subsection</span> </button> <ul id="toc-Special_cases-sublist" class="vector-toc-list"> <li id="toc-Affine_bundles_and_vector_bundles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Affine_bundles_and_vector_bundles"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Affine bundles and vector bundles</span> </div> </a> <ul id="toc-Affine_bundles_and_vector_bundles-sublist" class="vector-toc-list"> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Principal_bundles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Principal_bundles"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Principal bundles</span> </div> </a> <ul id="toc-Principal_bundles-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> 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href="/wiki/File:Solder_form_of_a_circle_over_a_torus.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Solder_form_of_a_circle_over_a_torus.jpg/220px-Solder_form_of_a_circle_over_a_torus.jpg" decoding="async" width="220" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Solder_form_of_a_circle_over_a_torus.jpg/330px-Solder_form_of_a_circle_over_a_torus.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Solder_form_of_a_circle_over_a_torus.jpg/440px-Solder_form_of_a_circle_over_a_torus.jpg 2x" data-file-width="3366" data-file-height="2167" /></a><figcaption>Solder form of a circle over a torus.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, more precisely in <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, a <b>soldering</b> (or sometimes <b>solder form</b>) of a <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundle</a> to a <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a> is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitively, soldering expresses in abstract terms the idea that a manifold may have a point of <a href="/wiki/Contact_(mathematics)" title="Contact (mathematics)">contact</a> with a certain model <a href="/wiki/Klein_geometry" title="Klein geometry">Klein geometry</a> at each point. In extrinsic differential geometry, the soldering is simply expressed by the tangency of the model space to the manifold. In intrinsic geometry, other techniques are needed to express it. Soldering was introduced in this general form by <a href="/wiki/Charles_Ehresmann" title="Charles Ehresmann">Charles Ehresmann</a> in 1950.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Soldering_of_a_fibre_bundle">Soldering of a fibre bundle</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solder_form&action=edit&section=1" title="Edit section: Soldering of a fibre bundle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>M</i> be a smooth manifold, and <i>G</i> a <a href="/wiki/Lie_group" title="Lie group">Lie group</a>, and let <i>E</i> be a smooth fibre bundle over <i>M</i> with structure group <i>G</i>. Suppose that <i>G</i> <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">acts transitively</a> on the typical fibre <i>F</i> of <i>E</i>, and that dim <i>F</i> = dim <i>M</i>. A <b>soldering</b> of <i>E</i> to <i>M</i> consists of the following data: </p> <ol><li>A distinguished <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">section</a> <i>o</i> : <i>M</i> → <i>E</i>.</li> <li>A linear isomorphism of vector bundles θ : T<i>M</i> → <i>o</i><sup>*</sup>V<i>E</i> from the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> of <i>M</i> to the <a href="/wiki/Pullback_bundle" title="Pullback bundle">pullback</a> of the <a href="/wiki/Vertical_bundle" class="mw-redirect" title="Vertical bundle">vertical bundle</a> of <i>E</i> along the distinguished section.</li></ol> <p>In particular, this latter condition can be interpreted as saying that θ determines a linear isomorphism </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{x}:T_{x}M\rightarrow V_{o(x)}E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>M</mi> <mo stretchy="false">→</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{x}:T_{x}M\rightarrow V_{o(x)}E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e7d4db399db5fb71632b1384de6e172c625ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.166ex; height:3.009ex;" alt="{\displaystyle \theta _{x}:T_{x}M\rightarrow V_{o(x)}E}"></span></dd></dl> <p>from the tangent space of <i>M</i> at <i>x</i> to the (vertical) tangent space of the fibre at the point determined by the distinguished section. The form θ is called the <b>solder form</b> for the soldering. </p> <div class="mw-heading mw-heading2"><h2 id="Special_cases">Special cases</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solder_form&action=edit&section=2" title="Edit section: Special cases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By convention, whenever the choice of soldering is unique or canonically determined, the solder form is called the canonical form, or the tautological form. </p> <div class="mw-heading mw-heading3"><h3 id="Affine_bundles_and_vector_bundles">Affine bundles and vector bundles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solder_form&action=edit&section=3" title="Edit section: Affine bundles and vector bundles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose that <i>E</i> is an affine <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundle</a> (a vector bundle without a choice of zero section). Then a soldering on <i>E</i> specifies first a <i>distinguished section</i>: that is, a choice of zero section <i>o</i>, so that <i>E</i> may be identified as a vector bundle. The solder form is then a linear isomorphism </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \colon TM\to V_{o}E,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>:</mo> <mi>T</mi> <mi>M</mi> <mo stretchy="false">→</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mi>E</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \colon TM\to V_{o}E,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f31a005dd490cd96ba6a13bb58472e5ef27b68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.624ex; height:2.509ex;" alt="{\displaystyle \theta \colon TM\to V_{o}E,}"></span></dd></dl> <p>However, for a vector bundle there is a canonical isomorphism between the vertical space at the origin and the fibre V<sub>o</sub><i>E</i> ≈ <i>E</i>. Making this identification, the solder form is specified by a linear isomorphism </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TM\to E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>M</mi> <mo stretchy="false">→</mo> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TM\to E.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/696b65fe376fd7feaa6c43f9cc521ddeecb3a7a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.115ex; height:2.176ex;" alt="{\displaystyle TM\to E.}"></span></dd></dl> <p>In other words, a soldering on an <a href="/wiki/Affine_bundle" title="Affine bundle">affine bundle</a> <i>E</i> is a choice of isomorphism of <i>E</i> with the tangent bundle of <i>M</i>. </p><p>Often one speaks of a <i>solder form on a vector bundle</i>, where it is understood <i>a priori</i> that the distinguished section of the soldering is the zero section of the bundle. In this case, the structure group of the vector bundle is often implicitly enlarged by the <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a> of <i>GL</i>(<i>n</i>) with the typical fibre of <i>E</i> (which is a representation of <i>GL</i>(<i>n</i>)).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Examples">Examples</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solder_form&action=edit&section=4" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>As a special case, for instance, the tangent bundle itself carries a canonical solder form, namely the identity.</li> <li>If <i>M</i> has a <a href="/wiki/Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metric</a> (or <a href="/wiki/Pseudo-Riemannian_metric" class="mw-redirect" title="Pseudo-Riemannian metric">pseudo-Riemannian metric</a>), then the <a href="/wiki/Covariant_metric_tensor" class="mw-redirect" title="Covariant metric tensor">covariant metric tensor</a> gives an isomorphism <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon TM\to T^{*}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>T</mi> <mi>M</mi> <mo stretchy="false">→</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\colon TM\to T^{*}M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8665d274e0fb1f3a7a899eb4d365bc6a68e5be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.059ex; height:2.676ex;" alt="{\displaystyle g\colon TM\to T^{*}M}"></span> from the tangent bundle to the <a href="/wiki/Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a>, which is a solder form.</li> <li>In <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a>, the solder form is known as the <a href="/wiki/Tautological_one-form" title="Tautological one-form">tautological one-form</a>, or alternately as the <b>Liouville one-form</b>, the <b>Poincaré one-form</b>, the <b>canonical one-form</b>, or the <b>symplectic potential</b>.</li> <li>Consider the Mobius strip as a fiber bundle over the circle. The vertical bundle <i>o</i><sup>*</sup>V<i>E</i> is still a Mobius strip, while the tangent bundle T<i>M</i> is the cylinder, so there is no solder form for this.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Applications">Applications</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solder_form&action=edit&section=5" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A solder form on a vector bundle allows one to define the <a href="/wiki/Torsion_tensor" title="Torsion tensor">torsion</a> and <a href="/wiki/Contorsion_tensor" title="Contorsion tensor">contorsion tensors</a> of a <a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">connection</a>.</li></ul> <ul><li>Solder forms occur in the <a href="/wiki/Sigma_model" title="Sigma model">sigma model</a>, where they glue together the tangent space of a spacetime manifold to the tangent space of the field manifold.</li></ul> <ul><li><a href="/wiki/Vierbein" class="mw-redirect" title="Vierbein">Vierbeins</a>, or <a href="/wiki/Tetrad_(general_relativity)" class="mw-redirect" title="Tetrad (general relativity)">tetrads</a> in general relativity, look like solder forms, in that they glue together coordinate charts on the spacetime manifold, to the preferred, usually orthonormal basis on the tangent space, where calculations can be considerably simplified. That is, the coordinate charts are the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea000afb5769206ddd5fd43f458430d04422ddeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.078ex; height:2.176ex;" alt="{\displaystyle TM}"></span> in the definitions above, and the frame field is the vertical bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle VE}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle VE}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51bf47cb33577128fa921fd2ad566aea3e16e001" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.563ex; height:2.176ex;" alt="{\displaystyle VE}"></span>. In the sigma model, the vierbeins are explicitly the solder forms.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Principal_bundles">Principal bundles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solder_form&action=edit&section=6" title="Edit section: Principal bundles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the language of principal bundles, a <b>solder form</b> on a smooth <a href="/wiki/Principal_bundle" title="Principal bundle">principal <i>G</i>-bundle</a> <i>P</i> over a <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a> <i>M</i> is a horizontal and <i>G</i>-equivariant <a href="/wiki/Vector-valued_differential_form" title="Vector-valued differential form">differential 1-form</a> on <i>P</i> with values in a <a href="/wiki/Linear_representation" class="mw-redirect" title="Linear representation">linear representation</a> <i>V</i> of <i>G</i> such that the associated <a href="/wiki/Bundle_map" title="Bundle map">bundle map</a> from the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> <i>TM</i> to the <a href="/wiki/Associated_bundle" title="Associated bundle">associated bundle</a> <i>P</i>×<sub><i>G</i></sub> <i>V</i> is a <a href="/wiki/Bundle_map" title="Bundle map">bundle isomorphism</a>. (In particular, <i>V</i> and <i>M</i> must have the same dimension.) </p><p>A motivating example of a solder form is the <a href="/wiki/Frame_bundle#Solder_form" title="Frame bundle">tautological or fundamental form</a> on the <a href="/wiki/Frame_bundle" title="Frame bundle">frame bundle</a> of a manifold. </p><p>The reason for the name is that a solder form solders (or attaches) the abstract principal bundle to the manifold <i>M</i> by identifying an associated bundle with the tangent bundle. Solder forms provide a method for studying <a href="/wiki/G-structure" class="mw-redirect" title="G-structure"><i>G</i>-structures</a> and are important in the theory of <a href="/wiki/Cartan_connection" title="Cartan connection">Cartan connections</a>. The terminology and approach is particularly popular in the physics literature. </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solder_form&action=edit&section=7" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Kobayashi (1957).</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Cf. Kobayashi (1957) section 11 for a discussion of the companion reduction of the structure group.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solder_form&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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="CITEREFEhresmann1950" class="citation journal cs1">Ehresmann, C. (1950). "Les connexions infinitésimales dans un espace fibré différentiel". <i>Colloque de Topologie, Bruxelles</i>: 29–55.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Colloque+de+Topologie%2C+Bruxelles&rft.atitle=Les+connexions+infinit%C3%A9simales+dans+un+espace+fibr%C3%A9+diff%C3%A9rentiel&rft.pages=29-55&rft.date=1950&rft.aulast=Ehresmann&rft.aufirst=C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASolder+form" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKobayashi1957" class="citation journal cs1">Kobayashi, Shoshichi (1957). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02411907">"Theory of Connections"</a>. <i>Ann. Mat. Pura Appl</i>. <b>43</b> (1): 119–194. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02411907">10.1007/BF02411907</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ann.+Mat.+Pura+Appl.&rft.atitle=Theory+of+Connections&rft.volume=43&rft.issue=1&rft.pages=119-194&rft.date=1957&rft_id=info%3Adoi%2F10.1007%2FBF02411907&rft.aulast=Kobayashi&rft.aufirst=Shoshichi&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252FBF02411907&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASolder+form" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKobayashi,_ShoshichiNomizu,_Katsumi1996" class="citation book cs1">Kobayashi, Shoshichi & Nomizu, Katsumi (1996). <i><a href="/wiki/Foundations_of_Differential_Geometry" title="Foundations of Differential Geometry">Foundations of Differential Geometry</a>, Vol. 1 & 2</i> (New ed.). <a href="/wiki/Wiley_Interscience" class="mw-redirect" title="Wiley Interscience">Wiley Interscience</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-15733-3" title="Special:BookSources/0-471-15733-3"><bdi>0-471-15733-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Differential+Geometry%2C+Vol.+1+%26+2&rft.edition=New&rft.pub=Wiley+Interscience&rft.date=1996&rft.isbn=0-471-15733-3&rft.au=Kobayashi%2C+Shoshichi&rft.au=Nomizu%2C+Katsumi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASolder+form" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" 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=Solder_form&oldid=1220996384">https://en.wikipedia.org/w/index.php?title=Solder_form&oldid=1220996384</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:Differential_forms" title="Category:Differential forms">Differential forms</a></li><li><a href="/wiki/Category:Fiber_bundles" title="Category:Fiber bundles">Fiber bundles</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 27 April 2024, at 05:58<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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