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invariant differential form

Definition

Invariant differential form

A differential form $\omega \in \Omega_{dR}^p(G)$ on a Lie group $G$ is called left invariant if for every $g \in G$ it is invariant under the pullback of differential forms

(1)$$(L_g)^* \omega = \omega$$

along the left multiplication action

$$\array{ L_g \colon & G &\longrightarrow& G \\ & x &\mapsto& g \cdot x }$$

Analogously a form is right invariant if it is invariant under the pullback by right translations $R_g$.

More generally, given a differentiable (e.g. smooth) group action of $G$ on a differentiable (e.g. smooth) manifold $M$

$$\array{ G \times M & \overset{\rho}{\longrightarrow} & M \\ (g,x) &\mapsto& g \cdot x }$$

then a differential form $\omega \in \Omega^p_{dR}(M)$ is called invariant if for all $g \in G$

$$\rho(g)^\ast(\omega) \;=\; \omega \,.$$

This reduces to the left invariance (1) for $M = G$ and $\rho$ being the left multiplication action of $G$ on itself.

Invariant vector field

For a vector field $X$ one instead typically defines the invariance via the pushforward $(T L_g) X = (L_g)_* X$. Regarding that $L_g$ and $T_g$ are diffeomorphisms, both pullbacks and pushfowards (hence invariance as well) are defined for every tensor field; and the two requirements are equivalent.

Related concepts

• invariant metric

• Maurer-Cartan form

• tangent Lie algebra

References

• page 89 (20 of 49) at MIT course on Lie groups (pdf 2)
• Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces
• F. Bruhat, Lectures on Lie groups and representations of locally compact groups, notes by S. Ramanan, TATA Bombay 1958, 1968, pdf
• MathOverflow: Construction of the Lie functor: left vs. right invariant vector fields on Lie groups and Lie groupoids