Algebraic and analytic properties of quasimetric spaces with dilations

TL;DR

This paper develops an axiomatic framework for understanding local tangent cones and differentiability in sub-Riemannian manifolds using dilation structures within quasimetric spaces, accommodating minimal smoothness assumptions.

Contribution

It introduces a dilation structure framework for quasimetric spaces, extending sub-Riemannian geometry theory beyond metric spaces with minimal smoothness requirements.

Findings

Established axiomatic approach for tangent cones in quasimetric spaces

Extended differentiability theory to sub-Riemannian manifolds with minimal smoothness

Showed that metric space theories do not directly apply to quasimetric spaces

Abstract

We provide an axiomatic approach to the theory of local tangent cones of regular sub-Riemannian manifolds and the differentiability of mappings between such spaces. This axiomatic approach relies on a notion of a dilation structure which is introduced in the general framework of quasimetric spaces. Considering quasimetrics allows us to cover a general case including, in particular, minimal smoothness assumptions on the vector fields defining the sub-Riemannian structure. It is important to note that the theory existing for metric spaces can not be directly extended to quasimetric spaces.