Cartan--Whitney Presentation, Non-smooth Analysis and Smoothability of Manifolds: On a theorem of Kondo--Tanaka

TL;DR

This paper provides a new proof connecting non-smooth analysis and manifold smoothability using geometric measure theory, extending results related to Lipschitz maps and exotic spheres.

Contribution

It introduces a simplified proof of a key theorem linking non-smooth analysis with manifold smoothability, utilizing tools from geometric measure theory.

Findings

New proof of Kondo--Tanaka theorem

Connections between non-smooth analysis and Whitney flat forms

Conditions for smoothability of homology manifolds

Abstract

Using tools and results from geometric measure theory, we give a simple new proof of the main result (Theorem 1.3) in K. Kondo and M. Tanaka, Approximation of Lipschitz Maps via Immersions and Differentiable Exotic Sphere Theorems, \textit{Nonlinear Anal.} \textbf{155} (2017), 219--249, as well as the converse statement. It explores the connections between the theory of non-smooth analysis {\it \`{a} la} F.~H. Clarke and the existence of special systems of Whitney flat $1$-forms with Sobolev regularity on certain families of homology manifolds.