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

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.

Typos corrected. This version is combined to arXiv:1910.07480. arXiv admin note: substantial text overlap with arXiv:1910.07480

References in corpus (3)

• Transport information geometry I: Riemannian calculus on probability simplex
• Log-Sobolev inequalities on the horizontal path space of a totally geodesic foliation
• Harnack inequalities on totally geodesic foliations with transverse Ricci flow

Cited by in corpus (2)

• Hypoelliptic entropy dissipation for stochastic differential equations
• The log-Sobolev inequality for spin systems of higher order interactions