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

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