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

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, $\mathit{\text{Nonlinear}}\mathit{\text{Anal.}}$ $\mathbf{\text{155}}$ (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.