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| In differential geometry, an '''equivariant differential form''' on a manifold ''M'' [[Lie group action|acted]] by a [[Lie group]] ''G'' is a [[polynomial map]] from the Lie algebra <math>\mathfrak{g} = \operatorname{Lie}(G)</math> to the space of differential forms on ''M'' that is equivariant; i.e., <math>\alpha(gX) = g\alpha(X)</math>. For an equivariant differential form <math>\alpha</math>, the '''equivariant exterior derivative''' <math>d_\mathfrak{g} \alpha</math> of <math>\alpha</math> is defined by
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| :<math>(d_\mathfrak{g} \alpha)(X) = d(\alpha(X)) - i_{X^\#}(\alpha(X))</math>
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| where ''d'' is the usual exterior derivative and <math>i_{X^\#}</math> is the interior product by the [[fundamental vector field]] generated by ''X''.
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| It is easy to see <math>d_\mathfrak{g} \circ d_\mathfrak{g} = 0</math> (use the fact the Lie derivative of <math>\alpha(X)</math> along <math>X^\#</math> is zero) and this makes the space of equivariant differential forms a complex. One can then put
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| :<math>H^*_G(X) = \operatorname{ker} d_\mathfrak{g}/\operatorname{im} d_\mathfrak{g}</math>,
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| which is called the [[equivariant cohomology]] of ''M''. The definition is due to H. Cartan. The notion has an application to the [[equivariant index theory]].
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| <math>d_\mathfrak{g}</math>-closed or <math>d_\mathfrak{g}</math>-exact forms are often called '''equivariantly closed''' or '''equivariantly exact'''.
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| == References ==
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| * {{Citation | last1=Berline | first1=Nicole | last2=Getzler | first2=E. | last3=Vergne | first3=Michèle | title=Heat Kernels and Dirac Operators | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=2004}}
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| [[Category:Differential geometry]]
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| {{geometry-stub}}
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Latest revision as of 01:15, 24 September 2014
22 year old Wood Machinist Golden from Deseronto, really loves blogging, property listing singapore developers in singapore and cave diving. Always loves planing a trip to destinations for example Kernave Archaeological Site (Cultural Reserve of Kernave).