On geometric quotients

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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.

Original languageEnglish (US)
Pages (from-to)3-26
Number of pages24
JournalAnnales Academiae Scientiarum Fennicae Mathematica
Issue number1
StatePublished - Dec 1 2009

All Science Journal Classification (ASJC) codes

  • General Mathematics


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