Exercise 27, § 6

Let G be a compact real Lie group, M a compact manifold of class $C^2$ and J an open interval of $\mathbf{R}$ containing 0. Let $(s, (x, \xi)) \mapsto (m_\xi(s, x), \xi)$ be a mapping of class $C^2$ of $G \times (M \times J)$ into $M \times J$ under which G operates on $M \times J$ on the left.

(a) Let X be a vector field of class $C^1$ on $M \times J$ such that, for all $(x, \xi) \in M \times J$, the second projection of $X_{(x, \xi)}$ is the tangent vector 1 to J. Translating X by G and integrating over G, deduce the existence of a field X' with the same properties as X, which is moreover invariant under G.

(b) Show that there exists a diffeomorphism $(x, \xi) \mapsto (h_\xi(x), \xi)$ of $M \times J$ onto itself such that (i) for all $\xi \in J$, $h_\xi$ is a diffeomorphism of M onto M; (ii) $m_\xi(s, x) = h_\xi(m_0(s, h_\xi^{-1}(x)))$ for $s \in G, x \in M, \xi \in J$. (Use (a) and Theorem 5 of no. 8.)