Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries

Let M be a smooth connected manifold endowed with a smooth measure mu and a smooth locally subelliptic diffusion operator L satisfying L1 = 0 and symmetric with respect to mu. Associated with L one has the carre du champ Gamma and a canonical distance d, with respect to which we suppose that (M, d) be complete. We assume that M is also equipped with another first-order differential bilinear form Gamma(Z) and we assume that Gamma and Gamma(Z) satisfy Hypotheses 1.1, 1.2, and 1.4 below. With these forms we introduce in (1.12) a generalization of the curvature-dimension inequality from Riemannian geometry (see Definition 1.3). In our main results we prove that, using solely (1.12), one can develop a theory which parallels the celebrated works of Yau and Li-Yau on complete manifolds with Ricci curvature bounded from below. We also obtain an analogue of the Bonnet-Myers theorem. In Section 2 we construct large classes of sub-Riemannian manifolds with transverse symmetries which satisfy the generalized curvature-dimension inequality (1.12). Such classes include all Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is bounded from below, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is bounded from below.