invariant differential form

1 Lie Groups

Let G be a Lie group and gG.

Let Lg:GG and Rg:GG be the functions of left and right multiplication by g (respectively). Let Cg:GG be the function of conjugationMathworldPlanetmath by g, i.e. Cg(h):=ghg-1.

A differential k-form ( ω on G is said to be

  • left invariant if Lg*ω=ω for every gG, where Lg* is the pullbackPlanetmathPlanetmath induced Lg.

  • right invariant if Rg*ω=ω for every gG, where Rg* is the pullback induced Rg.

  • invariant or bi-invariant if it is both left invariant and right invariant.

  • adjointMathworldPlanetmath invariant if Cg*ω=ω for every gG, where Cg* is the pullback induced by Cg.

Much like left invariant vector fields (, left invariant forms are uniquely determined by their values in Te(G), the tangent space at the identity elementMathworldPlanetmath eG, i.e. a left invairant form ω is uniquely determined by the values


This means that left invariant forms are uniquely determined by their values on the Lie algebra of G.

Under this setting, the space ΩLk(G) of left invariant k-forms can be identified with Hom(Λk𝔤,), the space of homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from the k-th exterior power of 𝔤 to , where 𝔤 denotes the Lie algebra of G.

- Let Ωk(G) be the space of k-forms in G. The exterior derivativeMathworldPlanetmath d:Ωk(G)Ωk+1(G) takes left invariant forms to left invariant forms. Moreover, the formulaMathworldPlanetmathPlanetmath for exterior derivative for left invariant forms simplifies to


where ωΩk(G) and X0,,Xk are left invariant vector fields in G.

Hence, the exterior derivative induces a map d:ΩLk(G)ΩLk+1(G) and (ΩL*(G),d) forms a chain complexMathworldPlanetmath. Thus, we can talk about the cohomology groups of left invariant forms.

Similar results hold for right invariant forms.

2 Manifolds

Suppose a Lie group G acts smoothly ( on a differential manifold M and let


denote the action of G.

A differential k-form ω in M is said to be invariant if tg*ω=ω for every gG, where tg* denotes the pullback induced by tg.

This definition reduces to the previous ones when we take M as the group G itself and when the action is

  • the action of G on itself by left multiplication.

  • the action of G on itself by right multiplication.

  • the action of G×G on G defined by t(g,h)(k):=gkh-1.

  • the action of G on itself by conjugation.

3 Compact Lie Group Actions

We now consider actions of a compact Lie group G on a manifold M. Let Ωk(M) the space of k-forms in M and ΩGk(M) the space of invariant k-forms in M. Let μ be the Haar measure of G.

From each k-form in M we can construct an invariant form by taking on its ””. Following this idea we define a map J:Ωk(M)ΩGk(M) by


where ωΩk(M) and X1,,Xk are vector fields of M.

The image of the map J is indeed in ΩGk(M) since for every hG:

th(J(ω))(X1,,Xk) = J(ω)((th)*X1,,(th)*Xk)
= 1μ(G)Gω((tg)*(th)*X1,,(tg)*(th)*Xk)𝑑μ(g)
= 1μ(G)Gω((tgh)*X1,,(tgh)*Xk)𝑑μ(g)
= 1μ(G)Gω((tg)*X1,,(tg)*Xk)𝑑μ(g)
= J(ω)(X1,,Xk)

Moreover, J is the identityPlanetmathPlanetmathPlanetmathPlanetmath for invariant k-forms. Suppose ωΩGk(M), then

J(ω)(X1,,Xk) = 1μ(G)Gtg*(ω)(X1,,Xk)𝑑μ(g)
= 1μ(G)Gω(X1,,Xk)𝑑μ(g)
= ω(X1,,Xk)

Theorem - The map J is a chain map, i.e. dJ=Jd, where d is the exterior derivative of a form.

From the previous observations we can see that the exterior derivative takes invariant forms to invariant forms, inducing a map d:ΩGk(M)ΩGk+1(M). Hence, (ΩG*(M),d) is a chain complex and we can talk about the cohomology groups of invariant forms in M.

4 Cohomology of Manifolds

Let G be a compact Lie group that acts smoothly on a manifold M (again, with the action denoted by tg).

Since tg is a diffeomorphism of M it induces an automorphism tg* on the cohomology groups Hk(M;). Hence, G acts as a group of automorphisms on Hk(M;). Let Hk(M;)G be the fixed pointPlanetmathPlanetmath set of this action.

Theorem - The inclusion I:ΩGk(M)Ωk(M) induces an isomorphismMathworldPlanetmathPlanetmath


If in G is connected, then tg and the identity 1M are homotopic, tg1M, for every gG. This implies that the induced automorphisms are the same, i.e. tg*=Id, where Id is the identity on Hk(M;). Hence, the fixed point set is the whole Hk(M;) and there is an isomorphism


Thus, the cohomology groups of a manifold where a compact connected Lie group acts are just the cohomology groups defined by the invariant forms on M. This means we can ”forget” the whole of differential forms in M and regard only those who are invariant.

Title invariant differential form
Canonical name InvariantDifferentialForm
Date of creation 2013-03-22 17:48:31
Last modified on 2013-03-22 17:48:31
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 25
Author asteroid (17536)
Entry type Definition
Classification msc 58A10
Classification msc 57T10
Classification msc 57S15
Classification msc 22E30
Classification msc 22E15
Synonym invariant form
Synonym bi-invariant form
Synonym bi-invariant differential form
Related topic CohomologyOfCompactConnectedLieGroups
Defines left invariant differential form
Defines left invariant form
Defines right invariant differential form
Defines right invariant form
Defines adjoint invariant form
Defines adjoint invariant differential form
Defines chain complex of invariant forms
Defines cohomologyMathworldPlanetmathPlanetmath of manifolds with a Lie group action