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

Abstract

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