A review of the tangent space in sub-Finsler geometry and applications to the failure of the $\mathsf{CD}$ condition

TL;DR

This paper reviews the tangent space construction in sub-Finsler geometry, explores its measure-theoretic properties, and demonstrates the failure of the curvature-dimension condition in certain 3D-contact sub-Finsler manifolds.

Contribution

It provides a detailed description of tangent spaces in sub-Finsler manifolds and applies this to analyze the curvature-dimension condition, revealing its failure in specific cases.

Findings

Tangent space in sub-Finsler geometry can be described via nilpotent approximation.

The curvature-dimension condition $ extsf{CD}$ fails in 3D-contact sub-Finsler manifolds.

The measure assumptions are crucial for tangent space characterization.

Abstract

We review the construction of the tangent space to a sub-Finsler manifold in the measured Gromov-Hausdorff sense. Under suitable assumptions on the measure, the metric measure tangent is described by the nilpotent approximation, equipped with a scalar multiple of the Lebesgue measure. We apply this result in the study of the Lott-Sturm-Villani curvature-dimension condition in sub-Finsler geometry. In particular, we show the failure of the $\mathsf{CD}$ condition in 3D-contact sub-Finsler manifolds, equipped with a bounded measure.