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href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="higher_geometry">Higher geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></strong> / <strong><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></strong></p> <p><strong>Ingredients</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></p> </li> </ul> <p><strong>Concepts</strong></p> <ul> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">little</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+scheme">generalized scheme</a></p> </li> </ul> </li> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">big</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-stacks">geometric ∞-stacks</a></li> </ul> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> / <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%28%E2%88%9E%2C1%29-site">étale (∞,1)-site</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-scheme">dg-scheme</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schematic+homotopy+type">schematic homotopy type</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived noncommutative geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></li> </ul> </li> <li> <p>derived smooth geometry</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Jones' theorem</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Deligne-Kontsevich conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+geometric+stacks">Tannaka duality for geometric stacks</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#InTermsOfSubbundlesOfTheFrameBundle'>In terms of subbundles of the frame bundle</a></li> <li><a href='#in_terms_of_cartan_connections'>In terms of Cartan connections</a></li> <li><a href='#in_higher_differential_geometry'>In higher differential geometry</a></li> <ul> <li><a href='#structure_on_a_principal_bundle'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure on a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-principal bundle</a></li> <li><a href='#structure_on_an_etale_groupoid'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-Structure on an etale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid</a></li> </ul> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#integrability_of_structure'>Integrability of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure</a></li> <li><a href='#relation_to_special_holonomy'>Relation to special holonomy</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#canonical_gstructures'>Canonical G-structures</a></li> <li><a href='#reduction_of_tangent_bundle_structure'>Reduction of tangent bundle structure</a></li> <li><a href='#lift_of_tangent_bundle_structure'>Lift of tangent bundle structure</a></li> <li><a href='#ExamplesOfReductionsOfNonTangentBundles'>Reduction of more general bundle structure</a></li> <li><a href='#complex_geometric_examples'>Complex geometric examples</a></li> <li><a href='#special_holonomy_examples'>Special holonomy examples</a></li> <li><a href='#structures_on_8manifolds'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-Structures on 8-manifolds</a></li> <li><a href='#higher_geometric_examples'>Higher geometric examples</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#traditional'>Traditional</a></li> <li><a href='#in_supergeometry'>In supergeometry</a></li> <li><a href='#ReferencesInSupergravity'>In supergravity</a></li> <li><a href='#in_complex_geometry'>In complex geometry</a></li> <li><a href='#in_higher_geometry'>In higher geometry</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A <strong>G-structure</strong> on an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, for a given <a class="existingWikiWord" href="/nlab/show/structure+group">structure group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-subbundle of the <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a> (of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>.</p> <p>Equivalently, this means that a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure is a choice of <a class="existingWikiWord" href="/nlab/show/reduction+of+structure+groups">reduction</a> of the canonical structure group <a class="existingWikiWord" href="/nlab/show/general+linear+group">GL(n)</a> of the <a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> to which the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> is <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated</a> along the given inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \hookrightarrow GL(n)</annotation></semantics></math>.</p> <p>More generally, one can consider the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is not a <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> but equipped with any group homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \to GL(n)</annotation></semantics></math>. If this is instead an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> one speaks of a <a class="existingWikiWord" href="/nlab/show/lift+of+structure+groups">lift of structure groups</a>.</p> <p>Both cases, in turn, can naturally be understood as special cases of <a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a>, which is a notion that applies more generally to <a class="existingWikiWord" href="/nlab/show/principal+infinity-bundles">principal infinity-bundles</a>.</p> <p>In this language, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure is a lift <em>in <a class="existingWikiWord" href="/nlab/show/classifying+stacks">classifying stacks</a></em> (<a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable</a> <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>) of the classifying map of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>G</mi><mi>structure</mi></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo maxsize="1.2em" minsize="1.2em">↓</mo> <mi>str</mi></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo>⊢</mo><mi>T</mi><mi>X</mi></mrow></munder></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{B}G \\ & {}^{ \mathllap{G structure} }\nearrow & \big\downarrow^{ str } \\ X &\underset{ \vdash T X }{\longrightarrow}& \mathbf{B} GL(n) } </annotation></semantics></math></div> <p>Beware the distinction to <em><a class="existingWikiWord" href="/nlab/show/tangential+structure">tangential structure</a></em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, where such a lift is considered (only) at the level of underlying <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a>.</p> <h2 id="definition">Definition</h2> <h3 id="general">General</h3> <p>Given a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> and given a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie</a> <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \hookrightarrow GL(n)</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a>, then a <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/reduction+of+structure+groups">reduction of the structure group</a> of the <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <p>There are many more explicit (and more abstract) equivalent ways to say this, which we discuss below. There are also many evident variants and generalizations.</p> <p>Notably one may consider reductions of the frames in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th order <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a>. (e. g. <a href="#Alekseevskii">Alekseevskii</a>) This yields order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure and the ordinary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structures above are then first order.</p> <p>Moreover, the definition makes sense for generalized <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> modeled on other base spaces than just <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a>. In particular there are evident generalizations to <a class="existingWikiWord" href="/nlab/show/supermanifolds">supermanifolds</a> and to <a class="existingWikiWord" href="/nlab/show/complex+manifolds">complex manifolds</a>.</p> <h3 id="InTermsOfSubbundlesOfTheFrameBundle">In terms of subbundles of the frame bundle</h3> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fr</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fr(X)</annotation></semantics></math>, and given a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>⟶</mo><mi>GL</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> G \longrightarrow GL(\mathbb{R}^n) </annotation></semantics></math></div> <p>into the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a>, then a <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> equipped with an inclusion of <a class="existingWikiWord" href="/nlab/show/fiber+bundles">fiber bundles</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd></mtd> <mtd><mo>↪</mo></mtd> <mtd></mtd> <mtd><mi>Fr</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P &&\hookrightarrow&& Fr(X) \\ & \searrow && \swarrow \\ && X } </annotation></semantics></math></div> <p>which is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-equivariant.</p> </div> <p>(<a href="#Sternberg64">Sternberg 64, section VII, def. 2.1</a>).</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>From this perspective, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure consists of the collection of all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/frame+field">frames</a> on a manifold. For instance for an <a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a> it consists of all those frames which are pointwise an <a class="existingWikiWord" href="/nlab/show/orthonormal+basis">orthonormal basis</a> of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> (with respect to the <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> which is defined by the orthonormal structure).</p> </div> <p>Accordingly:</p> <div class="num_defn" id="GStructureGeneratedByFrameField"> <h6 id="definition_3">Definition</h6> <p>Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \hookrightarrow GL(n)</annotation></semantics></math> and given any one <a class="existingWikiWord" href="/nlab/show/frame+field">frame field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Fr</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sigma \colon X \to Fr(X)</annotation></semantics></math> over a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, then acting with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> at each point produces a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-subbundle. This is called the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure <em>generated</em> by the frame field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>.</p> </div> <h3 id="in_terms_of_cartan_connections">In terms of Cartan connections</h3> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure equipped with compatible <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> data is equivalently a <a class="existingWikiWord" href="/nlab/show/Cartan+connection">Cartan connection</a> for the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>↪</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>⋊</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G \hookrightarrow \mathbb{R}^n \rtimes G)</annotation></semantics></math>.</p> <p>See at <em><a href="Cartan+connection#ExampleGStructures">Cartan connection – Examples – G-structures</a></em></p> <h3 id="in_higher_differential_geometry">In higher differential geometry</h3> <h4 id="structure_on_a_principal_bundle"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure on a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-principal bundle</h4> <p>We give an equivalent definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structures in terms of <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a> (“from the <a class="existingWikiWord" href="/nlab/show/nPOV">nPOV</a>”). This serves to clarify the slightly subtle but important difference between existence and choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure, and seamlessly embeds the notion into the more general context of <a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a>.</p> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">G \to K</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a>. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>K</mi></mrow><annotation encoding="application/x-tex"> \mathbf{c} : \mathbf{B}G \to \mathbf{B}K </annotation></semantics></math></div> <p>for the morphism of <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/Lie+groupoids">Lie groupoids</a> ( the <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth</a> <a class="existingWikiWord" href="/nlab/show/moduli+stacks">moduli stacks</a> of smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>- and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a>, respectively).</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> (or generally an <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> or <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>, etc.) Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>K</mi></mrow><annotation encoding="application/x-tex"> k \colon X \longrightarrow \mathbf{B}K </annotation></semantics></math></div> <p>be any choice of morphism <a class="existingWikiWord" href="/nlab/show/modulating+morphism">modulating</a> it.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X, \mathbf{B}G)</annotation></semantics></math> etc. for the <a class="existingWikiWord" href="/nlab/show/derived+hom+space">hom-groupoid</a> of <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth groupoids</a> / smooth stacks . This is equivalently the groupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal bundles over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and smooth <a class="existingWikiWord" href="/nlab/show/gauge+transformations">gauge transformations</a> between them.</p> <p>Then the <strong>groupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></strong> (with respect to the given morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">G \to K</annotation></semantics></math>) is the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle><msub><mi>Struc</mi> <mrow><mo stretchy="false">[</mo><mi>P</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>K</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">{</mo><mi>k</mi><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{c}Struc_{[P]}(X) := \mathbf{H}(X, \mathbf{B}G) \times_{\mathbf{H}(X, \mathbf{B}K)} \{k\} \,. </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>c</mi></mstyle><msub><mi>Struc</mi> <mrow><mo stretchy="false">[</mo><mi>P</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>k</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>K</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{c}Struc_{[P]}(X) &\longrightarrow& \ast \\ \downarrow & \swArrow_\simeq& \downarrow^{\mathrlap{k}} \\ \mathbf{H}(X, \mathbf{B}G) &\stackrel{\mathbf{H}(X, \mathbf{c})}{\longrightarrow}& \mathbf{H}(X, \mathbf{B}K) } </annotation></semantics></math></div> <p>(the groupoid of <em><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structure">twisted c-structures</a></em>).</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>If here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is trivial in that it factors through the point, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>K</mi></mrow><annotation encoding="application/x-tex">k \colon X \to \ast \to \mathbf{B}K</annotation></semantics></math> then this homotopy fiber product is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>K</mi><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X,K/G)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">K/G</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> (<a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a>) which itself sits in the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">/</mo><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>K</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K/G \to \mathbf{B}G \to \mathbf{B}K \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example">Example</h6> <p>Specifically, when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fr</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fr(X)</annotation></semantics></math> is <span class="newWikiWord">modulated<a href="/nlab/new/modulating+moprhism">?</a></span> by a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_X \colon X \to \mathbf{B} GL(n)</annotation></semantics></math> into the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> for the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>:</mo><mo>=</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K := GL(n)</annotation></semantics></math>. Then for any group homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \to GL(n)</annotation></semantics></math>, a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fr</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fr(X)</annotation></semantics></math>, as above.</p> </div> <h4 id="structure_on_an_etale_groupoid"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-Structure on an etale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid</h4> <p>We discuss the concept in the generality of <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>, formalized in <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a>.</p> <p>See at <em><a href="differential+cohesive+%28infinity%2C1%29-topos#structures">differential cohesion – G-Structure</a></em></p> <h2 id="properties">Properties</h2> <h3 id="integrability_of_structure">Integrability of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure</h3> <div class="num_defn" id="Integrability"> <h6 id="definition_5">Definition</h6> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure on a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is called <em>locally flat</em> (<a href="#Sternberg64">Sternberg 64, section VII, def. 24</a>) or <em>integrable</em> (e.g. <a href="#Alekseevskii">Alekseevskii</a>) if it is locally equivalent to the <em>standard flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure</em>, def. <a class="maruku-ref" href="#StandardFlatGStructure"></a>.</p> <p>This means that there is an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> by <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> of the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> such that the restriction of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure to each of these is equivalent to the standard flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure.</p> </div> <p>See at <em><a class="existingWikiWord" href="/nlab/show/integrability+of+G-structures">integrability of G-structures</a></em> for more on this</p> <p>The <a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a> to integrability of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure is the <em><a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></em>. See there for more.</p> <h3 id="relation_to_special_holonomy">Relation to special holonomy</h3> <p>The existence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structures on tangent bundles of <a class="existingWikiWord" href="/nlab/show/Riemannian+manifolds">Riemannian manifolds</a> is closely related to these having <a class="existingWikiWord" href="/nlab/show/special+holonomy">special holonomy</a>.</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,g)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a> <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/holonomy+group">holonomy group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hol</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hol(g) \subset O(n)</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>⊂</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \subset O(n)</annotation></semantics></math> some other <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,g)</annotation></semantics></math> admits a torsion-free G-structure precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hol</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hol(g)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/adjoint+action">conjugate</a> to a <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <p>Moreover, the space of such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structures is the <a class="existingWikiWord" href="/nlab/show/coset">coset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">G/L</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is the group of elements suchthat conjugating <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hol</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hol(g)</annotation></semantics></math> with them lands in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> </div> <p>This appears as (<a href="#Joyce">Joyce prop. 3.1.8</a>)</p> <h2 id="examples">Examples</h2> <h3 id="canonical_gstructures">Canonical G-structures</h3> <div class="num_defn" id="StandardFlatGStructure"> <h6 id="definition_6">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \hookrightarrow GL(n)</annotation></semantics></math> a subgroup, the <em>standard flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure</em> on the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure which is generated, via def. <a class="maruku-ref" href="#GStructureGeneratedByFrameField"></a>, from the canonical <a class="existingWikiWord" href="/nlab/show/frame+field">frame field</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> (the one which is the identity at each point, under the defining identifications).</p> </div> <p> <div class='num_remark' id='CanonicalHStructureOnFModH'> <h6>Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>⊂</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \subset G</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie</a> <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a>-inclusion, the canonical <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a> <a class="existingWikiWord" href="/nlab/show/coprojection">coprojection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G \to G/H</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> that exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>-structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math>.</p> </div> </p> <p>(e.g. <a href="#CapSlovak09">Čap-Slovak 09, p. 53</a>)</p> <h3 id="reduction_of_tangent_bundle_structure">Reduction of tangent bundle structure</h3> <ul> <li> <p>For the <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n, \mathbb{R})</annotation></semantics></math> of matrices of positive determinant, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℝ</mi><msup><mo stretchy="false">)</mo> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">GL(n, \mathbb{R})^+</annotation></semantics></math>-structure defines an <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>.</p> </li> <li> <p>For the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a>, an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math>-structure defines a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a>. (See the discussion at <a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a> and at</p> </li> <li> <p>For the <a class="existingWikiWord" href="/nlab/show/special+linear+group">special linear group</a>, an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SL(n,R)</annotation></semantics></math>-structure defines a <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a>.</p> </li> <li> <p>For the trivial group, an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>e</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{e\}</annotation></semantics></math>-structure consists of an <span class="newWikiWord">absolute parallelism<a href="/nlab/new/absolute+parallelism">?</a></span> of the manifold.</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn><mi>m</mi></mrow><annotation encoding="application/x-tex">n = 2 m</annotation></semantics></math> even, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(m, \mathbb{C})</annotation></semantics></math>-structure defines an <a class="existingWikiWord" href="/nlab/show/almost+complex+structure">almost complex structure</a> on the manifold. It must satisfy an integrability condition to be a <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a>.</p> </li> </ul> <h3 id="lift_of_tangent_bundle_structure">Lift of tangent bundle structure</h3> <p>An example for a lift of structure groups is</p> <ul> <li>for the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">spin(n)</annotation></semantics></math>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure is a <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>.</li> </ul> <p>This continues with lifts to the</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+group">string group</a> giving <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+group">fivebrane group</a> giving <a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a>.</p> </li> </ul> <h3 id="ExamplesOfReductionsOfNonTangentBundles">Reduction of more general bundle structure</h3> <ul> <li> <p>For general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">G \to K</annotation></semantics></math>, the corresponding notion of <a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> involves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-principal bundles (not necessarily underlying a tangent bundle).</p> </li> <li> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>O</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(n,n) \hookrightarrow O(2n,2n)</annotation></semantics></math>-structure is a <a class="existingWikiWord" href="/nlab/show/generalized+complex+structure">generalized complex structure</a>;</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>E</mi> <mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">H_n \to E_{n(n)}</annotation></semantics></math> the inclusion of the <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a> into the <a class="existingWikiWord" href="/nlab/show/split+real+form">split real form</a> of an <a class="existingWikiWord" href="/nlab/show/exceptional+Lie+group">exceptional Lie group</a>, the corresponding structure is an <a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a>.</p> </li> </ul> <h3 id="complex_geometric_examples">Complex geometric examples</h3> <ul> <li> <p>The choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(n, \mathbb{C})</annotation></semantics></math> as subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n, \mathbb{C})</annotation></semantics></math>, determines a complex Riemannian structure;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CO</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CO(n, \mathbb{C}) \hookrightarrow GL(n, \mathbb{C})</annotation></semantics></math>, a complex conformal structure;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(2n, \mathbb{C})\hookrightarrow GL(2n, \mathbb{C})</annotation></semantics></math>, an almost symplectic structure;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>n</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">GL(2, \mathbb{C}) GL(n, \mathbb{C}) \hookrightarrow GL(2n, \mathbb{C}), n \geq 3</annotation></semantics></math>, determines an almost quaternionic structure;</p> </li> <li> <p>more generally a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(m, \mathbb{C}) GL(n, \mathbb{C})</annotation></semantics></math>-structure on a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mi>n</mi></mrow><annotation encoding="application/x-tex">m n</annotation></semantics></math>-dimensional manifold is locally identical to a Grassmannian spinor structure.</p> </li> </ul> <h3 id="special_holonomy_examples">Special holonomy examples</h3> <div> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/normed+division+algebra">normed division algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mi>𝔸</mi><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\mathbb{A}\;</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math>Riemannian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔸</mi></mrow><annotation encoding="application/x-tex">\mathbb{A}</annotation></semantics></math>-manifolds<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math>special Riemannian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔸</mi></mrow><annotation encoding="application/x-tex">\mathbb{A}</annotation></semantics></math>-manifolds<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mi>ℝ</mi><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\mathbb{R}\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mi>ℂ</mi><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\mathbb{C}\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mi>ℍ</mi><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\mathbb{H}\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/quaternion-K%C3%A4hler+manifold">quaternion-Kähler manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/hyperk%C3%A4hler+manifold">hyperkähler manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/octonions">octonions</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mi>𝕆</mi><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\mathbb{O}\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Spin%287%29-manifold">Spin(7)-manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td></tr> </tbody></table> <p>(<a href="special+holonomy#Leung02">Leung 02</a>)</p> </div> <h3 id="structures_on_8manifolds"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-Structures on 8-manifolds</h3> <p>For discussion of <a class="existingWikiWord" href="/nlab/show/G-structures">G-structures</a> on <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed</a> <a class="existingWikiWord" href="/nlab/show/8-manifolds">8-manifolds</a> see <a href="8-manifold#GStructuresOn8Manifolds">there</a>.</p> <h3 id="higher_geometric_examples">Higher geometric examples</h3> <p>See the list at <a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structure">twisted differential c-structure</a>.</p> <p><br /></p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/structure+group">structure group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reduction+of+structure+groups">reduction of structure groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+holonomy">special holonomy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrability+of+G-structures">integrability of G-structures</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homothety">homothety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangential+structure">tangential structure</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="traditional">Traditional</h3> <p>The concept originates around the work of <a class="existingWikiWord" href="/nlab/show/Eli+Cartan">Eli Cartan</a> (<a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a>) and</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Shiing-Shen+Chern">Shiing-Shen Chern</a>, <em>The geometry of G-structures</em>, Bull. Amer. Math. Soc. 72(2): 167–219. 1966 (<a href="http://www.ams.org/journals/bull/1966-72-02/S0002-9904-1966-11473-8/S0002-9904-1966-11473-8.pdf">pdf</a>, <a href="http://projecteuclid.org/euclid.bams/1183527777">Euclid</a>)</li> </ul> <p>Textbook accounts:</p> <ul> <li id="Sternberg64"> <p><a class="existingWikiWord" href="/nlab/show/Shlomo+Sternberg">Shlomo Sternberg</a>, chapter VII of: <em>Lectures on differential geometry</em>, Prentice-Hall (1964), AMS (1983) [ISBNJ:978-0-8218-1385-0, <a href="https://bookstore.ams.org/chel-316">ams:chel-316</a>, <a href="https://archive.org/details/lecturesondiffer0000ster">ark:/13960/t1pg9dv6k</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Shoshichi+Kobayashi">Shoshichi Kobayashi</a>, <em>Transformation Groups in Differential Geometry</em> 1972, reprinted as: Classics in Mathematics Vol. 70, Springer 1995 (<a href="https://link.springer.com/book/10.1007/978-3-642-61981-6">doi:10.1007/978-3-642-61981-6</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Molino">Pierre Molino</a>, <em>Theorie des G-Structures: Le Probleme d’Equivalence</em>, Lecture Notes in Mathematics, Springer (1977) (<a href="https://www.springer.com/de/book/9783540082460">ISBN:978-3-540-37360-5</a>)</p> </li> <li id="CapSlovak09"> <p><a class="existingWikiWord" href="/nlab/show/Andreas+%C4%8Cap">Andreas Čap</a>, <a class="existingWikiWord" href="/nlab/show/Jan+Slov%C3%A1k">Jan Slovák</a>, Chapter 1 of: <em>Parabolic Geometries I – Background and General Theory</em>, AMS 2009 (<a href="http://bookstore.ams.org/surv-154">ISBN:978-1-4704-1381-1</a>)</p> </li> </ul> <p>See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Pierre+Molino">Pierre Molino</a>, <em>Sur quelques propriétés des G-structures</em>, J. Differential Geom. Volume 7, Number 3-4 (1972), 489-518 (<a href="https://projecteuclid.org/euclid.jdg/1214431168">euclid:jdg/1214431168</a>)</li> </ul> <p>Lecture notes:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Marius+Crainic">Marius Crainic</a>, Chapters 3 and 4 of: <em><a href="https://webspace.science.uu.nl/~crain101/DG-2015/">Differential geometry course</a></em>, 2015 (<a href="https://webspace.science.uu.nl/~crain101/DG-2015/main10.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/CrainicDifferentialGeometry15.pdf" title="pdf">pdf</a>)</p> </li> <li id="Pasquotto"> <p><a class="existingWikiWord" href="/nlab/show/Federica+Pasquotto">Federica Pasquotto</a>, <em>Linear <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structures by examples</em> (<a href="http://www.few.vu.nl/~pasquott/course16.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/PasquottoGStructures.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>Surveys include</p> <ul> <li id="Alekseevskii"><a class="existingWikiWord" href="/nlab/show/Dmitry+Vladimirovich+Alekseevsky">Dmitry Vladimirovich Alekseevsky</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure on a manifold</em> in M. Hazewinkel (ed.) <em>Encyclopedia of Mathematics, Volume 4</em> (<a href="https://encyclopediaofmath.org/wiki/G-structure">eom:G-structure</a>)</li> </ul> <p>See also</p> <ul> <li>Wikipedia, <em><a href="http://en.wikipedia.org/wiki/G-structure">G-structure</a></em></li> </ul> <p>Discussion with an eye towards <a class="existingWikiWord" href="/nlab/show/special+holonomy">special holonomy</a> is in</p> <ul> <li id="Joyce"><a class="existingWikiWord" href="/nlab/show/Dominic+Joyce">Dominic Joyce</a>, section 2.6 of <em>Compact manifolds with special holonomy</em> , Oxford Mathematical Monogrophs (200)</li> </ul> <p>Discussion with an eye towards <a class="existingWikiWord" href="/nlab/show/torsion+constraints+in+supergravity">torsion constraints in supergravity</a> is in</p> <ul> <li id="Lott90"><a class="existingWikiWord" href="/nlab/show/John+Lott">John Lott</a>, <em>The Geometry of Supergravity Torsion Constraints</em>, Comm. Math. Phys. 133 (1990), 563–615, (exposition in <a href="http://arxiv.org/abs/math/0108125">arXiv:0108125</a>)</li> </ul> <p>Discussion of G-structures more generally on <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a>:</p> <ul> <li> <p>A. V. Bagaev, N. I. Zhukova, <em>The Automorphism Groups of Finite Type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-Structures on Orbifolds</em>, Siberian Mathematical Journal 44, 213–224 (2003) (<a href="https://doi.org/10.1023/A:1022920417785">doi:10.1023/A:1022920417785</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Robert+Wolak">Robert Wolak</a>, <em>Orbifolds, geometric structures and foliations. Applications to harmonic maps</em>, Rendiconti del seminario matematico - Universita politecnico di Torino vol. 73/1 , 3-4 (2016), 173-187 (<a href="https://arxiv.org/abs/1605.04190">arXiv:1605.04190</a>)</p> </li> <li> <p>Sebastian Daza, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-Structures on Orbifolds</em> (2021) [<a href="https://www.teses.usp.br/teses/disponiveis/45/45131/tde-24092021-130626/publico/GstructuresOnOrbifolds.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Daza-OrbiGStructures.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>On <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><msub><mi>Br</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">G = Br_\infty</annotation></semantics></math> the infinite <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Frederick+R.+Cohen">Frederick R. Cohen</a>, <em>Braid orientations and bundles with flat connections</em>, Inventiones mathematicae <strong>46</strong> (1978) 99–110 [<a href="https://doi.org/10.1007/BF01393249">doi:10.1007/BF01393249</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jonathan+Beardsley">Jonathan Beardsley</a>, <em>On Braids and Cobordism Theories</em>, Glasgow (2022) [notes: <a href="https://www.jonathanbeardsley.com/GlasgowNotes2022.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Beardsley-BraidsAndCobordism.pdf" title="pdf">pdf</a>]</p> </li> </ul> <h3 id="in_supergeometry">In supergeometry</h3> <p>Discussion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structures in <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> includes</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dmitri+Alekseevsky">Dmitri Alekseevsky</a>, <a class="existingWikiWord" href="/nlab/show/Vicente+Cort%C3%A9s">Vicente Cortés</a>, <a class="existingWikiWord" href="/nlab/show/Chandrashekar+Devchand">Chandrashekar Devchand</a>, <a class="existingWikiWord" href="/nlab/show/Uwe+Semmelmann">Uwe Semmelmann</a>, <em>Killing spinors are Killing vector fields in Riemannian Supergeometry</em> (<a href="http://arxiv.org/abs/dg-ga/9704002">arXiv:dg-ga/9704002</a>)</p> <p>(on <a class="existingWikiWord" href="/nlab/show/Killing+spinors">Killing spinors</a> as super-<a class="existingWikiWord" href="/nlab/show/Killing+vectors">Killing vectors</a>)</p> </li> </ul> <h3 id="ReferencesInSupergravity">In supergravity</h3> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/G-structures">G-structures</a> in <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> and <a class="existingWikiWord" href="/nlab/show/superstring+theory">superstring theory</a>:</p> <p>In relation to <a class="existingWikiWord" href="/nlab/show/torsion+constraints+in+supergravity">torsion constraints in supergravity</a>:</p> <ul> <li id="Lott90"><a class="existingWikiWord" href="/nlab/show/John+Lott">John Lott</a>, <em>The Geometry of Supergravity Torsion Constraints</em>, Comm. Math. Phys. 133 (1990), 563–615, (exposition in <a href="http://arxiv.org/abs/math/0108125">arXiv:0108125</a>)</li> </ul> <p>As a way of speaking about <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+structure">Calabi-Yau structure</a> and <a class="existingWikiWord" href="/nlab/show/generalized+Calabi-Yau+structure">generalized Calabi-Yau structure</a>:</p> <ul> <li id="Prins16"><a class="existingWikiWord" href="/nlab/show/Dani%C3%ABl+Prins">Daniël Prins</a>, <em>On flux vacua, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(n)</annotation></semantics></math>-structures and generalised complex geometry</em>, Université Claude Bernard – Lyon I, 2015. (<a href="https://arxiv.org/abs/1602.05415">arXiv:1602.05415</a>, <a href="https://tel.archives-ouvertes.fr/tel-01280717">tel:01280717</a>)</li> </ul> <p>In relation to <a class="existingWikiWord" href="/nlab/show/BPS+states">BPS states</a>/partial reduction of <a class="existingWikiWord" href="/nlab/show/number+of+supersymmetries">number of supersymmetries</a> under <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Koerber">Paul Koerber</a>, <em>Lectures on Generalized Complex Geometry for Physicists</em>, Fortsch. Phys. 59: 169-242, 2011 (<a href="https://arxiv.org/abs/1006.1536">arXiv:1006.1536</a>)</p> <p>(with application to <a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jerome+Gauntlett">Jerome Gauntlett</a>, Dario Martelli, Stathis Pakis, <a class="existingWikiWord" href="/nlab/show/Daniel+Waldram">Daniel Waldram</a>, <em>G-Structures and Wrapped NS5-branes</em>, Commun.Math.Phys. 247 (2004) 421-445 (<a href="https://arxiv.org/abs/hep-th/0205050">arxiv:hep-th/0205050</a>)</p> <p>(application to <a class="existingWikiWord" href="/nlab/show/flux+compactifications">flux compactifications</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/J%C3%A9r%C3%B4me+Gaillard">Jérôme Gaillard</a>, <em>On <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structures in gauge/string duality</em>, 2011 (<a href="https://cronfa.swan.ac.uk/Record/cronfa42569">cronfa:42569</a> <a href="http://inspirehep.net/record/1340775">spire:1340775</a>, <a class="existingWikiWord" href="/nlab/files/GaillardGStructure.pdf" title="pdf">pdf</a>)</p> <p>(with application to <a class="existingWikiWord" href="/nlab/show/holographic+QCD">holographic QCD</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ulf+Danielsson">Ulf Danielsson</a>, <a class="existingWikiWord" href="/nlab/show/Giuseppe+Dibitetto">Giuseppe Dibitetto</a>, Adolfo Guarino, <em>KK-monopoles and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structures in M-theory/type IIA reductions</em>, JHEP 1502 (2015) 096 (<a href="https://arxiv.org/abs/1411.0575">arXiv:1411.0575</a>)</p> <p>(with application to <a class="existingWikiWord" href="/nlab/show/D6-branes">D6-branes</a>/<a class="existingWikiWord" href="/nlab/show/KK-monopoles">KK-monopoles</a> in <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ruben+Minasian">Ruben Minasian</a>, <a class="existingWikiWord" href="/nlab/show/Dani%C3%ABl+Prins">Daniël Prins</a>, <a class="existingWikiWord" href="/nlab/show/Hagen+Triendl">Hagen Triendl</a>, <em>Supersymmetric branes and instantons on curved spaces</em>, JHEP 10 (2017) 159 (<a href="https://arxiv.org/abs/1707.07002">arXiv:1707.07002</a>)</p> </li> </ul> <p>and specifically so for <a class="existingWikiWord" href="/nlab/show/M-theory+on+8-manifolds">M-theory on 8-manifolds</a>:</p> <ul> <li id="IshamPope88"> <p><a class="existingWikiWord" href="/nlab/show/Chris+Isham">Chris Isham</a>, <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, <em>Nowhere Vanishing Spinors and Topological Obstructions to the Equivalence of the NSR and GS Superstrings</em>, Class. Quant. Grav. 5 (1988) 257 (<a href="http://inspirehep.net/record/251240">spire:251240</a>, <a href="https://doi.org/10.1088/0264-9381/5/2/006">doi:10.1088/0264-9381/5/2/006</a>)</p> <p>(focus on <a class="existingWikiWord" href="/nlab/show/Spin%287%29-structure">Spin(7)-structure</a>)</p> </li> <li id="IshamPopeWarner88"> <p><a class="existingWikiWord" href="/nlab/show/Chris+Isham">Chris Isham</a>, <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+Warner">Nicholas Warner</a>, <em>Nowhere-vanishing spinors and triality rotations in 8-manifolds</em>, Classical and Quantum Gravity, Volume 5, Number 10, 1988 (<a href="http://cds.cern.ch/record/185144">cds:185144</a>, <a href="https://iopscience.iop.org/article/10.1088/0264-9381/5/10/009">doi:10.1088/0264-9381/5/10/009</a>)</p> <p>(focus on <a class="existingWikiWord" href="/nlab/show/Spin%287%29-structure">Spin(7)-structure</a>)</p> </li> <li> <p>Cezar Condeescu, <a class="existingWikiWord" href="/nlab/show/Andrei+Micu">Andrei Micu</a>, <a class="existingWikiWord" href="/nlab/show/Eran+Palti">Eran Palti</a>, <em>M-theory Compactifications to Three Dimensions with M2-brane Potentials</em>, JHEP 04 (2014) 026 (<a href="https://arxiv.org/abs/1311.5901">arxiv:1311.5901</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dani%C3%ABl+Prins">Daniël Prins</a>, <a class="existingWikiWord" href="/nlab/show/Dimitrios+Tsimpis">Dimitrios Tsimpis</a>, <em>IIA supergravity and M-theory on manifolds with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(4)</annotation></semantics></math> structure</em>, Phys. Rev. D 89.064030 (<a href="https://arxiv.org/abs/1312.1692">arXiv:1312.1692</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Elena+Babalic">Elena Babalic</a>, <a class="existingWikiWord" href="/nlab/show/Calin+Lazaroiu">Calin Lazaroiu</a>, <em>Singular foliations for M-theory compactification</em>, JHEP 03 (2015) 116 (<a href="https://arxiv.org/abs/1411.3497">arXiv:1411.3497</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Elena+Babalic">Elena Babalic</a>, <a class="existingWikiWord" href="/nlab/show/Calin+Lazaroiu">Calin Lazaroiu</a>, <em>Foliated eight-manifolds for M-theory compactification</em>, JHEP 01 (2015) 140 (<a href="https://arxiv.org/abs/1411.3148">arXiv:1411.3148</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Carlos+S.+Shahbazi">Carlos S. Shahbazi</a>, <em>M-theory on non-Kähler manifolds</em>, JHEP 09 (2015) 178 [<a href="https://arxiv.org/abs/1503.00733">arXiv:1503.00733</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Elena+Babalic">Elena Babalic</a>, <a class="existingWikiWord" href="/nlab/show/Calin+Lazaroiu">Calin Lazaroiu</a>, <em>The landscape of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structures in eight-manifold compactifications of M-theory</em>, JHEP 11 (2015) 007 (<a href="https://arxiv.org/abs/1505.02270">arXiv:1505.02270</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Elena+Babalic">Elena Babalic</a>, <a class="existingWikiWord" href="/nlab/show/Calin+Lazaroiu">Calin Lazaroiu</a>, <em>Internal circle uplifts, transversality and stratified <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structures</em>, JHEP 11 (2015) 174 (<a href="https://arxiv.org/abs/1505.05238">arXiv:1505.05238</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Robin+Terrisse">Robin Terrisse</a>, <a class="existingWikiWord" href="/nlab/show/Dimitrios+Tsimpis">Dimitrios Tsimpis</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(3)</annotation></semantics></math> structures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math> bundles over four-manifolds</em>, JHEP09 (2017) 133 (<a href="https://arxiv.org/abs/1707.04636">arXiv:1707.04636</a>)</p> </li> <li> <p>Vikas Yadav, Aalok Misra, <em>On M-Theory Dual of Large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> Thermal QCD-Like Theories up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><msup><mi>R</mi> <mn>4</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(R^4)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-Structure Classification of Underlying Non-Supersymmetric Geometries</em> (<a href="https://arxiv.org/abs/2004.07259">arXiv:2004.07259</a>)</p> <blockquote> <p>(relation to <a class="existingWikiWord" href="/nlab/show/holographic+QCD">holographic QCD</a>)</p> </blockquote> </li> <li> <p>Mateo Galdeano, <em>The Geometry and Superconformal Algebras of String Compactifications with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure</em> [<a href="https://arxiv.org/abs/2212.13781">arXiv:2212.13781</a>]</p> <blockquote> <p>(relation to <a class="existingWikiWord" href="/nlab/show/superconformal+algebras">superconformal algebras</a>)</p> </blockquote> </li> </ul> <p>See also</p> <ul> <li> <p>Johannes Held, Section 3 of: <em>Non-SupersymmetricFlux Compactifications of Heterotic String- and M-theory</em> (<a href="https://edoc.ub.uni-muenchen.de/14341/1/Held_Johannes.pdf">pdf</a>)</p> </li> <li id="Papadopoulos18"> <p><a class="existingWikiWord" href="/nlab/show/George+Papadopoulos">George Papadopoulos</a>, <em>Geometry and symmetries of null <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structures</em> (<a href="https://arxiv.org/abs/1811.03500">arXiv:1811.03500</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Waldram">Daniel Waldram</a>, <em>Fluxes, holography and uses of Exceptional Generalized Geometry</em>, talk at <em><a class="existingWikiWord" href="/nlab/show/Strings+2022">Strings 2022</a></em> [<a href="https://indico.cern.ch/event/1085701/contributions/4940880">indico:4940880</a>, <a href="https://indico.cern.ch/event/1085701/contributions/4940880/attachments/2482886/4262576/Daniel%20Waldram%20Strings2022-waldram.pdf">slides</a>, <a href="https://ustream.univie.ac.at/media/core.html?id=e222a3a9-ebf4-445c-ada0-f7b1d592d908">video</a> ]</p> <blockquote> <p>on generalized <a class="existingWikiWord" href="/nlab/show/G-structures">G-structures</a> and <a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a> of <a class="existingWikiWord" href="/nlab/show/flux+compactifications">flux compactifications</a> and in <a class="existingWikiWord" href="/nlab/show/AdS%2FCFT">AdS/CFT</a></p> </blockquote> </li> </ul> <h3 id="in_complex_geometry">In complex geometry</h3> <ul> <li>Sergey Merkulov, <em>On group theoretic aspects of the non-linear twistor transform</em>, (<a href="http://people.maths.ox.ac.uk/lmason/Tn/40/TN40-07.pdf">pdf</a>)</li> </ul> <p>and his chapter A in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yuri+Manin">Yuri Manin</a>, <em>Gauge Field Theory and Complex Geometry</em>, Springer.</p> </li> <li> <p>Norman Wildberger, <em>On the complexication of the classical geometries and exceptional numbers</em>, (<a href="http://web.maths.unsw.edu.au/~norman/papers/L.pdf">pdf</a>)</p> </li> <li> <p>Jun-Muk Hwang, <em>Rational curves and prolongations of G-structures</em>,<a href="https://arxiv.org/abs/1703.03160">arXiv:1703.03160</a></p> </li> </ul> <h3 id="in_higher_geometry">In higher geometry</h3> <p>Some discussion in <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a> is in section 4.4.2 of</p> <ul> <li><em><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></em></li> </ul> <p>Formalization in <a class="existingWikiWord" href="/nlab/show/modal+type+theory">modal</a> <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> is in</p> <ul> <li id="Wellen17"> <p><a class="existingWikiWord" href="/nlab/show/Felix+Wellen">Felix Wellen</a>, <em><a class="existingWikiWord" href="/schreiber/show/thesis+Wellen">Formalizing Cartan Geometry in Modal Homotopy Type Theory</a></em>, 2017</p> </li> <li id="Wellen18"> <p><a class="existingWikiWord" href="/nlab/show/Felix+Wellen">Felix Wellen</a>, <em><a class="existingWikiWord" href="/schreiber/show/Cartan+Geometry+in+Modal+Homotopy+Type+Theory">Cartan Geometry in Modal Homotopy Type Theory</a></em> (<a href="https://arxiv.org/abs/1806.05966">arXiv:1806.05966</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 25, 2024 at 11:36:59. See the <a href="/nlab/history/G-structure" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/G-structure" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/3132/#Item_28">Discuss</a><span class="backintime"><a href="/nlab/revision/G-structure/55" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/G-structure" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/G-structure" accesskey="S" class="navlink" id="history" rel="nofollow">History (55 revisions)</a> <a href="/nlab/show/G-structure/cite" style="color: black">Cite</a> <a href="/nlab/print/G-structure" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/G-structure" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>