TY - JOUR

T1 - Approximations of Lipschitz maps via Ehresmann’s fibrations and Reeb’s sphere theorem for Lipschitz functions

AU - Kondo, Kei

N1 - Publisher Copyright:
Copyright © 2018, The Authors. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2018/11/10

Y1 - 2018/11/10

N2 - We show, as our main theorem, that if a Lipschitz map from a compact Riemannian manifold M to a connected compact Riemannian manifold N, where dim M ≥ dim N, has no singular points on M in the sense of F.H. Clarke, then the map admits a smooth approximation via Ehresmann’s fibrations. We also show the Reeb sphere theorem for a closed Riemannian manifold M of m:= dim M ≥ 2 which admits a Lipschitz function F: M −→ R with only two singular points, denoted by z1, z2 ∈ M, in the sense of Clarke by assuming that there is a constant c between F (z1) and F (z2) such that F −1(c) is homeomorphic to an (m − 1)-dimensional sphere. In the proof of our sphere theorem we use a corollary arising from the process of the proof of the main theorem above.Primary 49J52, 53C20, Secondary 57R12, 57R55

AB - We show, as our main theorem, that if a Lipschitz map from a compact Riemannian manifold M to a connected compact Riemannian manifold N, where dim M ≥ dim N, has no singular points on M in the sense of F.H. Clarke, then the map admits a smooth approximation via Ehresmann’s fibrations. We also show the Reeb sphere theorem for a closed Riemannian manifold M of m:= dim M ≥ 2 which admits a Lipschitz function F: M −→ R with only two singular points, denoted by z1, z2 ∈ M, in the sense of Clarke by assuming that there is a constant c between F (z1) and F (z2) such that F −1(c) is homeomorphic to an (m − 1)-dimensional sphere. In the proof of our sphere theorem we use a corollary arising from the process of the proof of the main theorem above.Primary 49J52, 53C20, Secondary 57R12, 57R55

KW - Convex analysis

KW - Ehresmann’s fibration

KW - Lipschitz map

KW - Nonsmooth analysis

KW - Reeb’s sphere theorem

KW - Smooth approximation

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M3 - Article

AN - SCOPUS:85093311301

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SN - 0402-1215

ER -