Universal non-CD of sub-Riemannian manifolds

Abstract We prove that a sub-Riemannian manifold with a full-support Radon measure is never CD ⁡ ( K , N ) \operatorname{CD}(K,N) for any K ∈ R K\in\mathbb{R} and N ∈ ( 1 , ∞ ) N\in(1,\infty) unless it is Riemannian. This generalizes previous non- CD \operatorname{CD} results for sub-Riemannian manifolds, where a measure with smooth and positive density is considered. Our proof is based on the analysis of the tangent cones and the geodesics within. Secondly, we construct new RCD \operatorname{RCD} structures on R n \mathbb{R}^{n} , named cone-Grushin spaces, that fail to be sub-Riemannian due to the lack of a scalar product along a curve, yet exhibit characteristic features of sub-Riemannian geometry, such as horizontal directions, large Hausdorff dimension, and inhomogeneous metric dilations.