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

@article{Feng2020SubRiemannianRC, title={Sub-Riemannian Ricci curvature via generalized Gamma \$z\$ calculus}, author={Qi Feng and Wuchen Li}, journal={arXiv: Differential Geometry}, year={2020} }

We derive sub-Riemannian Ricci curvature tensor for sub-Riemannian manifolds. We provide examples including the Heisenberg group, displacement group ($\textbf{SE}(2)$), 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. Our derivation of Ricci curvature is based on generalized Gamma $z$ calculus and $z$--Bochner's formula, where $z$ stands for extra directions…

## 2 Citations

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We generalize the Gamma $z$ calculus to study degenerate drift-diffusion processes, where $z$ stands for extra directions introduced into the degenerate system. Based on this calculus, we establish…

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. We study the Lyapunov convergence analysis for degenerate and non-reversible stochastic diﬀerential equations (SDEs). We apply the Lyapunov method to the Fokker-Planck equation, in which the…

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