Universal non-CD of sub-Riemannian manifolds

Abstract We prove that a sub-Riemannian manifold with a full-support Radon measure is never <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>CD</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>K</m:mi> <m:mo>,</m:mo> <m:mi>N</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{CD}(K,N) for any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>K</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> </m:math> K\in\mathbb{R} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>N</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">∞</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> N\in(1,\infty) unless it is Riemannian. This generalizes previous non- <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>CD</m:mi> </m:math> \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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>RCD</m:mi> </m:math> \operatorname{RCD} structures on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> \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.