Generalized Bochner formulas and Ricci lower bounds for sub-Riemannian manifolds of rank two

Abstract

We study a new 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. Sub-Riemannian manifolds for which this inequality is satisfied are shown to share many properties in common with Riemannian manifolds whose Ricci curvature is bounded from below

71 pages

References in corpus (1)

• Subelliptic Li-Yau estimates on three dimensional model spaces

Cited by in corpus (7)

• Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds
• Some properties of sub-Laplaceans
• Harnack inequalities on totally geodesic foliations with transverse Ricci flow
• The Li-Yau Inequality and Heat Kernels on Metric Measure Spaces
• Bishop and Laplacian Comparison Theorems on Three Dimensional Contact Subriemannian Manifolds with Symmetry
• Connections and Curvature in sub-Riemannian geometry
• Perelman's entropy and doubling property on Riemannian manifolds