Let $M$ be a Hilbert Riemannian manifold and $G$ a finite-dimensional Lie group acting on $M$. It is well-known that when $M$, too, is finite-dimensional $$\exp_p: U \subset T_pM \rightarrow \exp_p(U) \subset M $$ is a diffeomorphism for a small set $U$. We can also assume that it is $G$-equivariant if we chose the Riemannian metric on $M$ so that $G$ acts by isometries.
Now, the question is:
What can go wrong in the above case, when $M$ is a Hilbert manifold? (Note that $G$ should always be finite-dimensional)
See also Remark B.27 here.
