Exercise 4, § 1

Let H be a compact real Lie group operating on a finite-dimensional real manifold V; suppose that V is of class $C^r$, where $r \in \mathbf{N}_\mathbf{R}$, and that the action of H on V is of class $C^r$. Let $p \in V$ be invariant under H; the group H operates linearly on the tangent space T to V at $p$. Show that there exist an open neighbourhood U of $p$ which is stable under H and a $C^r$-morphism $f : U \to T$ such that: (a) $f(p) = 0$ and the tangent mapping to $f$ at $p$ is the identity; (b) $f$ commutes with the action of H on U and T. (Choose first an $f_0$ satisfying (a) and then define $f$ by the formula

where $dh$ is the Haar measure on H with total mass 1.)

Deduce that the H-spaces V and T are locally isomorphic to a neighbourhood of $p$, in other words that there exist systems of coordinates on V at $p$ with respect to which H operates linearly ("Bochner's Theorem").