Transverse Weitzenb\"ock formulas and curvature dimension inequalities   on Riemannian foliations with totally geodesic leaves
  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. 
Fabrice Baudoin, Bumsik Kim, Jing Wang
math.DG math.AP math.PR