Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries

Abstract

We prove that, on any sub-Riemannian manifold endowed with a positive smooth measure, the Bakry-\'Emery inequality for the corresponding sub-Laplacian implies the existence of enough Killing vector fields on the tangent cone to force the latter to be Euclidean at each point, yielding the failure of the curvature-dimension condition in full generality. Our approach does not apply to non-strictly-positive measures. In fact, we prove that the weighted Grushin plane does not satisfy any curvature-dimension condition, but, nevertheless, does admit an a.e. pointwise version of the Bakry-\'Emery inequality. As recently observed by Pan and Montgomery, one half of the weighted Grushin plane satisfies the RCD(0,N) condition, yielding a counterexample to gluing theorems in the RCD setting.