. Measure Contraction Properties of Contact Sub-riemannian Manifolds with Symmetry

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 contact sub-Riemannian manifold with a one-parameter family of symmetries to satisfy these properties. Moreover, in the special case where the quotient of the contact subRiemannian manifold by the symmetries is Kähler, the sufficient conditions are defined by a combination of the holomorphic sectional curvature and the Ricci curvature of the quotient manifold. This generalizes the earlier work in [2] for the three dimensional case and in [14] 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 [17, 18]. The crucial new tool here is a certain decoupling of the corresponding matrix Riccati equation. It is important to note that our measure contraction properties for the considered class of sub-Riemannian structures cannot be improved because the corresponding inequalities turn to be equalities for the corresponding homogeneous models. Using the same scheme, we also prove a version of Bonnet-Myer’s Theorem in our setting.