Definition 9

Let $G$ be a Lie group, $M$ a manifold of class $C^r$ and $f$ a mapping of class $C^r$ of $M$ into $G$. The left (resp. right) differential of $f$ is the differential form of degree 1 on $M$ with values in $L(G)$ which associates with every vector $u \in T_m(M)$ the element $f(m)^{-1} \cdot (T_m f)(u)$ (resp. $(T_m f)(u) \cdot f(m)^{-1}$).

In this chapter we shall only consider the left differential, which we shall denote by $f^{-1}.df$, and leave to the reader the task of translating the results for the right differential.

If $f$ is the identity mapping of $G$, $f^{-1}.df$ is the canonical left differential form $\omega$ of $G$. Returning to the general case of Definition 8,

and hence $f^{-1}.df = f^*(\omega)$. This implies that $f^{-1}.df$ is of class $C^{r-1}$.