• Corpus ID: 214802512

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… 
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References

SHOWING 1-10 OF 30 REFERENCES
Generalized Gamma $z$ calculus via sub-Riemannian density manifold
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
Bakry–Émery curvature and model spaces in sub-Riemannian geometry
We prove comparison theorems for the sub-Riemannian distortion coefficients appearing in interpolation inequalities. These results, which are equivalent to a sub-Laplacian comparison theorem for the
Sub-Laplacians on Sub-Riemannian Manifolds
We consider different sub-Laplacians on a sub-Riemannian manifold M. Namely, we compare different natural choices for such operators, and give conditions under which they coincide. One of these
Curvature Dimension Inequalities and Subelliptic Heat Kernel Gradient Bounds on Contact Manifolds
AbstractWe study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain: Geometric conditions ensuring
A Formula for Popp’s Volume in Sub-Riemannian Geometry
Abstract For an equiregular sub-Riemannian manifold M, Popp’s volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the
Transport information geometry: Riemannian calculus on probability simplex
We formulate the Riemannian calculus of the probability set embedded with $$L^2$$ L 2 -Wasserstein metric. This is an initial work of transport information geometry. Our investigation starts with the
Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries
Let $\M$ be a smooth connected manifold endowed with a smooth measure $\mu$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $\mu$.
A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality
Let $$\mathbb M $$M be a smooth connected manifold endowed with a smooth measure $$\mu $$μ and a smooth locally subelliptic diffusion operator $$L$$L satisfying $$L1=0$$L1=0, and which is symmetric
Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations
We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical
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