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

The Lott-Sturm-Villani curvature-dimension condition C D(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 C D(K, N) condition, for any K E R and N E (1, 8). However, his result does not cover the case of almost-Riemannian manifolds. In this paper, we address the problem of disproving the C D condition in this setting, providing a new strategy which allows us to contradict the one-dimensional version of the C D condition. In particular, we prove that 2-dimensional almost-Riemannian manifolds and strongly regular almost-Riemannian manifolds do not satisfy the C D(K, N) condition for any K E R and N E (1, 8).