On geometric quotients

The following results are established: i) Let f: M→H be a C¹ map of a compact connected C¹ manifold (without boundary) into a Hilbert space. Then the map f is a C¹ fibre bundle projection onto f(M) if and only if f^-1: f(M)→H (M) is Lipschitz. Here, H (M) denotes the metric space of nonempty closed subsets of M with the Hausdorff metric. ii) Let M and N be compact connected C1 manifolds (without boundary) and let f: M→N be a C¹ map. Then f is a Lipschitz fibre bundle projection if and only if it is a C¹ fibre bundle projection. iii) Let G × M→M be a C¹ action of a compact Lie group on a compact connected C¹ manifold (without boundary) and let f: M→H be an invariant C¹ map. Then the map f induces a bi-Lipschitz embedding of M=G (with respect to the quotient metric) into H if and only if f induces a C¹ embedding of M=G (with respect to the C¹ quotient structure) into H. Moreover, in contrast to the result of Schwarz in the C^∞ case, such an embedding f exists exactly when the action has a single orbit type.