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

Abstract

Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces. We consider Sasakian manifolds with non-negative Tanaka-Webster Ricci curvature equipped with the metric measure space structure defined by the sub-Riemannian metric and the Popp measure. We show that these spaces satisfy the measure contraction property M CP (0, N ) for some positive integer N . We also show that the same result holds when the Sasakian manifold is equipped with a family of Riemannian metrics extending the sub-Riemannian one. Sub-Riemannian Manifolds and the Measure Contraction PropertyIn this section, we recall various notions and facts about contact sub-Riemannian manifolds and the measure contraction property.A sub-Riemannian manifold is a manifold M equipped with a distribution D and a positive definite metric ·, · SR defined on D. A smooth 8 , Y 2 )V = 0, (4) Rm(X 1 , V )X 2 = 0,

Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

“…Inspired by some of the results in [28] and [36], we prove in this section that for Sasakian manifolds, a comparison theorem for the sub-Riemannian distance may be obtained as a limit when ε → 0 of a comparison theorem for the distances r ε . With respect to [2,28,30], we obtain an explicit and simple upper bound for ∆ H r ε which is sharp when ε → 0, and in our opinion the method and computations are more straightforward and shorter. Our method has also the advantage to easily yield a Hessian comparison theorem for the distance r ε , ε > 0 (such Hessian comparison theorem is not explicitly worked out in [30]) and a vertical Laplacian comparison theorem (see Theorem 3.9).…”

“…Such a Π is called a dynamic optimal coupling from δ x 0 to ν. In our case, we have a measurable map G ε : M → Geo ε (M) so that each G ε (x) is a minimal g ε geodesic from x 0 to x by a measurable selection theorem (the existence of such map is classical when ε > 0 and we refer to [31] in the case ε = 0). Then, the push-forward measure G ♯ ν indeed provides a dynamic optimal coupling from δ x 0 to ν.…”

“…The first approach is more intrinsic and yields curvature invariants from the sub-Riemannian structure only. Though it gives a deep understanding of the geodesics and, in principle, provides a general framework, it is somehow challenging to compute and to make use of those invariants, even in simple examples like Sasakian spaces (see [5,30,31]). The second approach is more extrinsic and produces curvature quantities from the sub-Riemannian structure together with the choice of a natural complement to the horizontal distribution.…”

Volume and distance comparison theorems for sub-Riemannian manifolds

“…as proven in [3,4,34] in increasing degrees of generality, in terms of a measure contraction property essentially equivalent to (1.2). Notice that the aforementioned class of structures includes the Heisenberg groups and the Hopf fibrations, for which (1.2) is sharp.…”

On Measure Contraction Property without Ricci Curvature Lower Bound

“…Note also that, under the same assumptions as in Theorem 1.1, it was shown in [10] that (M, d CC , vol P ) satisfies MCP (0, 2n + 3), where d CC is the Carnot-Caratheordory distance and vol P is the Popp measure (see also [8,1] for the earlier results).…”