Annales Academić Scientiarum Fennicć
Mathematica
Volumen 34, 2009, 3-26

ON GEOMETRIC QUOTIENTS

Hossein Movahedi-Lankarani and Robert Wells

Penn State Altoona, Department of Mathematics and Statistics
Altoona, PA 16601-3760, U.S.A.; hxm9 'at' psu.edu

Penn State University, Department of Mathematics
University Park, PA 16802, U.S.A.; wells 'at' math.psu.edu

Abstract. The following results are established:
i) Let f:M\to 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)\to 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\to 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 x M\to M be a C¹ action of a compact Lie group on a compact connected C¹ manifold (without boundary) and let f:M\to 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^\infty case, such an embedding f exists exactly when the action has a single orbit type.

2000 Mathematics Subject Classification: Primary 58C25, 54E35, 57R40.

Key words: Manifold, quotient, smooth structure, metric structure, Lipschitz, Lie group, smooth action, spherically compact, invariant polynomial.

Reference to this article: H. Movahedi-Lankarani and R. Wells: On geometric quotients. Ann. Acad. Sci. Fenn. Math. 34 (2009), 3-26.

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