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 -