Any sub-Riemannian Metric has Points of Smoothness

TL;DR

This paper proves that any sub-Riemannian metric has points of smoothness and that the metric is analytic on a dense subset in complete real-analytic cases, advancing understanding of the geometric structure.

Contribution

It establishes the existence of regular points in the sub-Riemannian exponential map and shows analyticity on a dense subset for real-analytic manifolds.

Findings

Existence of regular points in the sub-Riemannian exponential map

Analyticity of the metric on a dense subset in real-analytic cases

Advances understanding of sub-Riemannian geometric structure

Abstract

We prove the result stated in the title; it is equivalent to the existence of a regular point of the sub-Riemannian exponential mapping. We also prove that the metric is analytic on an open everywhere dense subset in the case of a complete real-analytic sub-Riemannian manifold.