Communications in Analysis and Geometry
Volume 24 (2016)
Transverse Weitzenböck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves
Pages: 913 – 937
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