# 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, 2016, 36 (1) : 303-321. doi: 10.3934/dcds.2016.36.303
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##### References:
 [1] João Paulo da Silva, Julio López, Ricardo Dahab. Isogeny formulas for Jacobi intersection and twisted hessian curves. Advances in Mathematics of Communications, 2020, 14 (3) : 507-523. doi: 10.3934/amc.2020048 [2] Martins Bruveris. Completeness properties of Sobolev metrics on the space of curves. Journal of Geometric Mechanics, 2015, 7 (2) : 125-150. doi: 10.3934/jgm.2015.7.125 [3] Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2021, 13 (1) : 55-72. doi: 10.3934/jgm.2020031 [4] Farid Tari. Geometric properties of the integral curves of an implicit differential equation. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 349-364. doi: 10.3934/dcds.2007.17.349 [5] Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223 [6] Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159 [7] Nicolas Dirr, Federica Dragoni, Max von Renesse. Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach. Communications on Pure & Applied Analysis, 2010, 9 (2) : 307-326. doi: 10.3934/cpaa.2010.9.307 [8] Kai Zhao, Wei Cheng. On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4345-4358. doi: 10.3934/dcds.2019176 [9] Qing Liu, Atsushi Nakayasu. Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 157-183. doi: 10.3934/dcds.2019007 [10] Erlend Grong, Alexander Vasil’ev. Sub-Riemannian and sub-Lorentzian geometry on $SU(1,1)$ and on its universal cover. Journal of Geometric Mechanics, 2011, 3 (2) : 225-260. doi: 10.3934/jgm.2011.3.225 [11] J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford and Masahico Saito. State-sum invariants of knotted curves and surfaces from quandle cohomology. Electronic Research Announcements, 1999, 5: 146-156. [12] Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121 [13] Daniel Genin, Serge Tabachnikov. On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards. Journal of Modern Dynamics, 2007, 1 (2) : 155-173. doi: 10.3934/jmd.2007.1.155 [14] Yunlong Huang, P. S. Krishnaprasad. Sub-Riemannian geometry and finite time thermodynamics Part 1: The stochastic oscillator. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1243-1268. doi: 10.3934/dcdss.2020072 [15] Stefan Sommer, Anne Marie Svane. Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 391-410. doi: 10.3934/jgm.2017015 [16] Beatrice Abbondanza, Stefano Biagi. Riesz-type representation formulas for subharmonic functions in sub-Riemannian settings. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021101 [17] Alice B. Tumpach, Stephen C. Preston. Quotient elastic metrics on the manifold of arc-length parameterized plane curves. Journal of Geometric Mechanics, 2017, 9 (2) : 227-256. doi: 10.3934/jgm.2017010 [18] Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295 [19] Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327 [20] Lei Zhang, Anfu Zhu, Aiguo Wu, Lingling Lv. Parametric solutions to the regulator-conjugate matrix equations. Journal of Industrial & Management Optimization, 2017, 13 (2) : 623-631. doi: 10.3934/jimo.2016036

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