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

We prove that, on any sub-Riemannian manifold endowed with a positive smooth measure, the Bakry-emery inequality for the corresponding sub-Laplacian, 1 2 & UDelta;( backward difference u 2) & GE; g( backward difference u, backward difference & UDelta;u) + K  backward difference u 2, K & ISIN; R, 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. & COPY; 2023 Elsevier Inc. All rights reserved.

Automated summary

We prove that the Bakry-emery inequality for the corresponding sub-Laplacian implies the failure of the curvature-dimension condition on sub-Riemannian manifolds. Our approach does not apply to non-strictly positive measures. We also show that the weighted Grushin plane does not satisfy any curvature-dimension condition but admits a pointwise version of the Bakry-emery inequality.