Sub-riemannian geometry from intrinsic viewpoint

TL;DR

This paper develops an intrinsic, axiomatic approach to sub-Riemannian geometry using dilation structures, integrating Gromov's metric perspective with Siebert's algebraic framework without relying on differential structures.

Contribution

It introduces approximately 12 axioms to describe sub-Riemannian geometry purely through metric and dilation structures, avoiding traditional differential assumptions.

Findings

Sub-Riemannian spaces can be characterized by dilation structures.

The approach unifies Gromov's metric and Siebert's algebraic perspectives.

Provides a differential-structure-free axiomatic framework.

Abstract

Gromov proposed to extract the (differential) geometric content of a sub-riemannian space exclusively from its Carnot-Carath\'eodory distance. One of the most striking features of a regular sub-riemannian space is that it has at any point a metric tangent space with the algebraic structure of a Carnot group, hence a homogeneous Lie group. Siebert characterizes homogeneous Lie groups as locally compact groups admitting a contracting and continuous one-parameter group of automorphisms. Siebert result has not a metric character. In these notes I show that sub-riemannian geometry may be described by about 12 axioms, without using any a priori given differential structure, but using dilation structures instead. Dilation structures bring forth the other intrinsic ingredient, namely the dilations, thus blending Gromov metric point of view with Siebert algebraic one.