Transverse Weitzenb\"ock formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves

TL;DR

This paper develops new Weitzenb"ock formulas for Riemannian foliations with totally geodesic leaves, leading to curvature dimension inequalities that imply Li-Yau estimates and a sub-Riemannian Bonnet-Myers theorem based on intrinsic horizontal geometry.

Contribution

It introduces a family of Weitzenb"ock formulas parametrized by metric variations, establishing curvature dimension inequalities in this geometric setting.

Findings

Derived new Weitzenb"ock formulas for Riemannian foliations.

Established curvature dimension inequalities under natural conditions.

Proved Li-Yau estimates and a sub-Riemannian Bonnet-Myers theorem.

Abstract

We prove a family of new Weitzenb\"ock formulas on a Riemannian foliation with totally geodesic leaves. These Weitzenb\"ock 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.