Ricci curvaturetype lower bounds for sub-Riemannian structures on Sasakian manifolds

Paul W. Y.  Lee; Chengbo  Li; Igor  Zelenko

doi:10.3934/dcds.2016.36.303

Article Contents

2016, Volume 36, Issue 1: 303-321. Doi: 10.3934/dcds.2016.36.303

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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
2.
Department of Mathematics, Tianjin University, Tianjin, 300072

Received: March 31, 2014

Revised: February 28, 2015

Published: May 2017

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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 [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.

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Mathematics Subject Classification: Primary: 53C17, 34C10, 53C25; Secondary: 53D10, 70G45, 53C55.

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