Atlas as solution of Sincov's inequality

TL;DR

This paper presents a solution to Sincov's inequality and extends the concept of an atlas to non-differentiable and discontinuous cases by interpreting solutions as differentiable manifolds.

Contribution

It introduces a novel approach to solving Sincov's inequality and generalizes the notion of an atlas beyond differentiable manifolds.

Findings

Solution of Sincov's inequality found

Atlas concept extended to non-differentiable cases

Manifold interpretation in the differentiable case

Abstract

We find a solution of Sincov's inequality. Further, we prove that in the differentiable case we can interpret such solution as a differentiable manifold in the original sense of Lang. This allows to generalize the notion of atlas and transition map for non-differentiable and discontinuous case.