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

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 $N\in(1,\infty)$.