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# Communications in Analysis and Geometry

## Volume 24 (2016)

### Number 5

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

Pages: 913 – 937

DOI: http://dx.doi.org/10.4310/CAG.2016.v24.n5.a1

#### Authors

#### Abstract

We prove a family of 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, sharp eigenvalue estimates and a sub-Riemannian Bonnet–Myers compactness theorem whose assumptions only rely on the intrinsic geometry of the horizontal distribution.

#### Keywords

sub-Riemannian geometry, Weitzenböck formula, Bochner method, Riemannian foliation

#### 2010 Mathematics Subject Classification

53C12, 53C17

Published 6 March 2017