Almost-Riemannian manifolds do not satisfy the curvature-dimension condition

Abstract

The Lott–Sturm–Villani curvature-dimension condition textsf{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 (Rev Mat Iberoam 37(1), 177–188, 2021) that a large class of sub-Riemannian manifolds do not satisfy the textsf{CD}(K,N) condition, for any Kin {mathbb {R}} and Nin (1,infty ). However, his result does not cover the case of almost-Riemannian manifolds. In this paper, we address the problem of disproving the textsf{CD} condition in this setting, providing a new strategy which allows us to contradict the one-dimensional version of the textsf{CD} condition. In particular, we prove that 2-dimensional almost-Riemannian manifolds and strongly regular almost-Riemannian manifolds do not satisfy the textsf{CD}(K,N) condition for any Kin {mathbb {R}} and Nin (1,infty ).