"Heat kernel analysis and generalized Ricci lower bounds in
sub-Riemannian geometry"

 Fabrice Baudoin (Purdue Univ.)

We study a class of rank two sub-Riemannian manifolds encompassing Riemannian manifolds, CR manifolds with vanishing Webster-Tanaka torsion, orthonormal bundles over Riemannian manifolds, and graded nilpotent Lie groups of step two. These manifolds admit a canonical horizontal connection and a canonical sub-Laplacian. We construct on these manifolds an analogue of the Riemannian Ricci tensor and prove Bochner type formulas for the sub-Laplacian. As a consequence, it is possible to formulate on these spaces a sub-Riemannian analogue of the so-called curvature dimension inequality. The heat kernel analysis on sub-Riemannian manifolds for which this inequality is satisfied is shown to share many properties in common with the heat kernel analysis on Riemannian manifolds whose Ricci curvature is bounded from below. This is mainly a joint work with N. Garofalo.