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

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