Transverse Weitzenböck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves

Abstract

We prove a family of new Weitzenböck formulas on a Riemannian foliation with totally geodesic leaves. These Weitzenböck formulas are naturally parametrized by the canonical variation of the metric. As a consequence, under natural geometric conditions, the horizontal Laplacian satisfies a generalized curvature dimension inequality. Among other things, this curvature dimension inequality implies Li-Yau estimates for positive solutions of the horizontal heat equation and a sub-Riemannian Bonnet-Myers compactness theorem whose assumptions only rely on the intrinsic geometry of the horizontal distribution.

To be published in Communications in Analysis and Geometry

References in corpus (2)

• Sub-Laplacians on sub-Riemannian manifolds
• Stochastic analysis on sub-Riemannian manifolds with transverse symmetries

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• Log-Sobolev inequalities on the horizontal path space of a totally geodesic foliation
• Harnack inequalities on totally geodesic foliations with transverse Ricci flow
• Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations, Part II
• Convergence to equilibrium for hypoelliptic non-symmetric Ornstein-Uhlenbeck type operators
• Riemannian and Sub-Riemannian geodesic flows