Surface measure on, and the local geometry of, sub-Riemannian manifolds

TL;DR

This paper derives an integral formula for measuring hypersurfaces in equiregular sub-Riemannian manifolds and introduces criteria for distance convergence and local asymptotics of small metric balls.

Contribution

It provides a new integral formula for spherical measures and establishes general criteria for distance convergence and local geometric asymptotics in sub-Riemannian geometry.

Findings

Derived an integral formula for hypersurface measures

Established criteria for uniform convergence of sub-Riemannian distances

Analyzed local asymptotics of small metric balls

Abstract

We prove an integral formula for the spherical measure of hypersurfaces in equiregular sub-Riemannian manifolds. Among various technical tools, we establish a general criterion for the uniform convergence of parametrized sub-Riemannian distances, and local uniform asymptotics for the diameter of small metric balls.