TY - JOUR

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

AU - Kondo, Kei

N1 - Funding Information:
2010 Mathematics Subject Classification. Primary 53C20; Secondary 49J52, 57R12, 57R55. Key Words and Phrases. convex analysis, Ehresmann fibration, Lipschitz map, nonsmooth analysis, Reeb’s sphere theorem, smooth approximation. Supported by the Grant-in-Aid for Science Reserch (C), JSPS KAKENHI Grant Numbers 16K05133, 17K05220, 18K03280.
Publisher Copyright:
© 2022 The Mathematical Society of Japan

PY - 2022

Y1 - 2022

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 Clarke, then the map admits a smooth approximation via Ehresmann fibrations. We also show the Reeb sphere theorem for Lipschitz functions, i.e., if a closed Riemannian manifold admits a Lipschitz function with exactly two singular points in the sense of Clarke, then the manifold is homeomorphic to the sphere.

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 Clarke, then the map admits a smooth approximation via Ehresmann fibrations. We also show the Reeb sphere theorem for Lipschitz functions, i.e., if a closed Riemannian manifold admits a Lipschitz function with exactly two singular points in the sense of Clarke, then the manifold is homeomorphic to the sphere.

KW - convex analysis

KW - Ehresmann fibration

KW - Lipschitz map

KW - nonsmooth analysis

KW - Reeb’s sphere theorem

KW - smooth approximation

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U2 - 10.2969/JMSJ/83448344

DO - 10.2969/JMSJ/83448344

M3 - Article

AN - SCOPUS:85129982653

VL - 74

SP - 521

EP - 548

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 2

ER -