# American Institute of Mathematical Sciences

January  2016, 36(1): 303-321. doi: 10.3934/dcds.2016.36.303

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

 1 Room 216, Lady Shaw Building, The Chinese University of Hong Kong, Shatin, Hong Kong, China 2 Department of Mathematics, Tianjin University, Tianjin, 300072, China, China

Received  April 2014 Revised  March 2015 Published  June 2015

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 [2] for the three dimensional case and in [19] 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 [24,25]. 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.
Citation: Paul W. Y. Lee, Chengbo Li, Igor Zelenko. Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 303-321. doi: 10.3934/dcds.2016.36.303
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