### 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) |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Annales Academiae Scientiarum Fennicae Mathematica*,

*34*(1), 3-26.

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*Annales Academiae Scientiarum Fennicae Mathematica*, vol. 34, no. 1, pp. 3-26.

**On geometric quotients.** / Movahedi-Lankarani, Hossein; Wells, Robert.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On geometric quotients

AU - Movahedi-Lankarani, Hossein

AU - Wells, Robert

PY - 2009/12/1

Y1 - 2009/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77952557110&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77952557110&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:77952557110

VL - 34

SP - 3

EP - 26

JO - Annales Academiae Scientiarum Fennicae Mathematica

JF - Annales Academiae Scientiarum Fennicae Mathematica

SN - 1239-629X

IS - 1

ER -