### Abstract

The following results are established: i) Let f: M→H be a C^{1} map of a compact connected C^{1} manifold (without boundary) into a Hilbert space. Then the map f is a C^{1} 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 C^{1} map. Then f is a Lipschitz fibre bundle projection if and only if it is a C^{1} fibre bundle projection. iii) Let G × M→M be a C^{1} action of a compact Lie group on a compact connected C^{1} manifold (without boundary) and let f: M→H be an invariant C^{1} 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 C^{1} embedding of M=G (with respect to the C^{1} 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.

Original language | English (US) |
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Pages (from-to) | 3-26 |

Number of pages | 24 |

Journal | Annales Academiae Scientiarum Fennicae Mathematica |

Volume | 34 |

Issue number | 1 |

State | Published - Dec 1 2009 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Annales Academiae Scientiarum Fennicae Mathematica*,

*34*(1), 3-26.