Almost-Riemannian manifolds do not satisfy the $\mathsf{CD}$ condition

TL;DR

This paper demonstrates that 2-dimensional almost-Riemannian manifolds and strongly regular variants do not satisfy the curvature-dimension condition $ extsf{CD}(K,N)$, extending previous results to this specific class of geometric spaces.

Contribution

The paper introduces a new strategy to disprove the $ extsf{CD}$ condition in almost-Riemannian manifolds, filling a gap left by prior work.

Findings

Almost-Riemannian manifolds do not satisfy $ extsf{CD}(K,N)$ for any $K$ and $N$.

The new method contradicts the 1-dimensional $ extsf{CD}$ condition.

Results apply to 2D and strongly regular almost-Riemannian manifolds.

Abstract

The Lott-Sturm-Villani curvature-dimension condition $\mathsf{CD}(K,N)$ provides a synthetic notion for a metric-measure space to have curvature bounded from below by $K$ and dimension bounded from above by $N$. It was proved by Juillet that a large class of \sr manifolds do not satisfy the $\mathsf{CD}(K,N)$ condition, for any $K\in\mathbb R$ and $N\in(1,\infty)$. However, his result does not cover the case of almost-Riemannian manifolds. In this paper, we address the problem of disproving the $\mathsf{CD}$ condition in this setting, providing a new strategy which allows us to contradict the $1$-dimensional version of the $\mathsf{CD}$ condition. In particular, we prove that $2$-dimensional almost-Riemannian manifolds and strongly regular almost-Riemannian manifolds do not satisfy the $\mathsf{CD}(K,N)$ condition for any $K\in\mathbb R$ and $N\in(1,\infty)$.