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

TL;DR

This paper demonstrates that curvature-dimension conditions generally fail on sub-Riemannian manifolds with positive measures due to tangent cone isometries, providing counterexamples and clarifying limitations of Bakry-Émery inequalities.

Contribution

It shows that Bakry-Émery inequalities imply tangent cone Euclideanness, leading to failure of curvature-dimension conditions on sub-Riemannian manifolds, with specific examples like the weighted Grushin plane.

Findings

Bakry-Émery inequality implies tangent cone Euclideanness.

Weighted Grushin plane does not satisfy any curvature-dimension condition.

Weighted Grushin plane admits an a.e. pointwise Bakry-Émery inequality.

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.