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

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 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.