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arXiv · 2004.01863

Sub-Riemannian Ricci curvature via generalized Gamma $z$ calculus

Qi Feng·Wuchen Li
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Abstract

We derive sub-Riemannian Ricci curvature tensor for sub-Riemannian manifolds. We provide examples including the Heisenberg group, displacement group, and Martinet sub-Riemannian structure with arbitrary weighted volumes, in which we establish analytical bounds for sub-Riemannian curvature dimension bounds and log-Sobolev inequalities. {These bounds can be used to establish the entropy dissipation results for sub-Riemannian drift diffusion processes on a compact spatial domain, in term of $L_1$ distance.} Our derivation of Ricci curvature is based on generalized Gamma $z$ calculus and $z$--Bochner's formula, where $z$ stands for extra directions introduced into the sub-Riemannian degenerate structure.

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Qi Feng, Wuchen Li. 2020-04-04. Sub-Riemannian Ricci curvature via generalized Gamma $z$ calculus. https://arxiv.org/abs/2004.01863

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