Approximations of Lipschitz maps via Ehresmann fibrations and Reeb's sphere theorem for Lipschitz functions

2018/11/11 by Kondo, Kei
#53C20 #57R55 #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Topology (math.GT) #Optimization and Control (math.OC) #Primary 49J52 #Secondary 57R12

paper · doi:10.48550/arxiv.1811.04340

Abstract

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 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.

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