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



Fabrice Baudoin (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

Bumsik Kim (Department of Mathematics, University of Connecticut, Storrs, Ct., U.S.A.)

Jing Wang (IMA, University of Minnesota, Minneapolis, Minn., U.S.A.)


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.


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

2010 Mathematics Subject Classification

53C12, 53C17

Published 6 March 2017