Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds

TL;DR

This paper establishes measure contraction properties for Sasakian manifolds with sub-Riemannian distances, linking these properties to Tanaka-Webster curvature, and extends previous results to higher dimensions using advanced geometric analysis.

Contribution

It provides new sufficient conditions based on Tanaka-Webster curvature for measure contraction properties on Sasakian manifolds, generalizing earlier three-dimensional and Heisenberg group results.

Findings

Sufficient conditions for measure contraction properties derived from Tanaka-Webster curvature.

Extension of previous work to higher-dimensional Sasakian manifolds.

Exact formulas for measure contraction in homogeneous models.

Abstract

Measure contraction properties are generalizations of the notion of Ricci curvature lower bounds in Riemannian geometry to more general metric measure spaces. In this paper, we give sufficient conditions for a Sasakian manifold equipped with a natural sub-Riemannian distance to satisfy these properties. Moreover, the sufficient conditions are defined by the Tanaka-Webster curvature. This generalizes the earlier work in \cite{AgLe1} for the three dimensional case and in \cite{Ju} for the Heisenberg group. To obtain our results we use the intrinsic Jacobi equations along sub-Riemannian extremals, coming from the theory of canonical moving frames for curves in Lagrangian Grassmannians \cite{LiZe1, LiZe2}. The crucial new tool here is a certain decoupling of the corresponding matrix Riccati equation. It is also worth pointing out that our method leads to exact formulas for the measure contraction in the case of the corresponding homogeneous models in the considered class of sub-Riemannian structures.