## Abstract

As our main theorem, we prove that a Lipschitz map from a compact Riemannian manifold M into a Riemannian manifold N admits a smooth approximation via immersions if the map has no singular points on M in the sense of F.H. Clarke, where dimM≤dimN. As its corollary, we have that if a bi-Lipschitz homeomorphism between compact manifolds and its inverse map have no singular points in the same sense, then they are diffeomorphic. We have three applications of the main theorem: The first two of them are two differentiable sphere theorems for a pair of topological spheres including that of exotic ones. The third one is that a compact n-manifold M is a twisted sphere and there exists a bi-Lipschitz homeomorphism between M and the unit n-sphere S^{n}(1) which is a diffeomorphism except for a single point, if M satisfies certain two conditions with respect to critical points of its distance function in the Clarke sense. Moreover, we have three corollaries from the third theorem; the first one is that for any twisted sphere Σ^{n} of general dimension n, there exists a bi-Lipschitz homeomorphism between Σ^{n} and S^{n}(1) which is a diffeomorphism except for a single point. In particular, there exists such a map between an exotic n-sphere Σ^{n} of dimension n>4 and S^{n}(1); the second one is that if an exotic 4-sphere Σ^{4} exists, then Σ^{4} does not satisfy one of the two conditions above; the third one is that for any Grove–Shiohama type n-sphere N, there exists a bi-Lipschitz homeomorphism between N and S^{n}(1) which is a diffeomorphism except for one of points that attain their diameters.

Original language | English |
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Pages (from-to) | 219-249 |

Number of pages | 31 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 155 |

DOIs | |

Publication status | Published - May 1 2017 |

## Keywords

- Bi-Lipschitz homeomorphism
- Differentiable sphere theorem
- Exotic spheres
- Lipschitz map
- Non-smooth analysis
- Smooth approximation

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics