Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries

TL;DR

This paper generalizes curvature-dimension inequalities for sub-Riemannian manifolds with symmetries, enabling the development of geometric analysis tools similar to those in Riemannian geometry, and proves results like Li-Yau inequalities and Bonnet-Myers type theorems.

Contribution

It introduces a new generalized curvature-dimension inequality for sub-Riemannian manifolds with transverse symmetries and develops a parallel theory to classical Riemannian results.

Findings

Establishes a generalized curvature-dimension inequality for sub-Riemannian manifolds.

Proves Li-Yau type inequalities and Bonnet-Myers theorem under this framework.

Constructs classes of manifolds satisfying the new inequality, including Sasakian and Carnot groups.

Abstract

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 $\mu$. Associated with $L$ one has \textit{le carr\'e du champ} $\Gamma$ and a canonical distance $d$, with respect to which we suppose that $(M,d)$ be complete. We assume that $\M$ is also equipped with another first-order differential bilinear form $\Gamma^Z$ and we assume that $\Gamma$ and $\Gamma^Z$ satisfy the Hypothesis below. With these forms we introduce in \eqref{cdi} below a generalization of the curvature-dimension inequality from Riemannian geometry, see Definition \ref{D:cdi}. In our main results we prove that, using solely \eqref{cdi}, one can develop a theory which parallels the celebrated works of Yau, and Li-Yau on complete manifolds with Ricci bounded from below. We also obtain an analogue of the Bonnet-Myers theorem. In Section \ref{S:appendix} we construct large classes of sub-Riemannian manifolds with transverse symmetries which satisfy the generalized curvature-dimension inequality \eqref{cdi}. Such classes include all Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is bounded from below, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is bounded from below.