nLab
invariant differential form

Definition

Let $M$ be a differential manifold with differentiable left action of Lie group $G$, $G\times M\to M$ (respectively right action $M\times G\to G$). For example, the multiplication map of $G$ on itself. Then we define the left translations $L_g : m\mapsto g m$ (resp. right translations $R_g: m\mapsto m g$) for every $g\in G$, which are both diffeomorphisms of $M$.

A differential form on a Lie group $\omega \in \Omega^1(G)$ is called left invariant if for every $g \in G$ it is invariant under the pullback by the translation $L_g$

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

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

• 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