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

Let be a smooth connected manifold endowed with a smooth measure μ and a smooth locally subelliptic diffusion operator L satisfying $L1=0$, and which is symmetric with respect to μ. Associated with L one has carr\'e du champ Γ and a canonical distance d, with respect to which we suppose that $(M,d)$ be complete. We assume that is also equipped with another first-order differential bilinear form ΓZ and we assume that Γ and ΓZ satisfy the Hypothesis below. With these forms we introduce in below a generalization of the curvature-dimension inequality from Riemannian geometry, see Definition :cdi. In our main results we prove that, using solely , one can develop a theory which parallels the celebrated works of Yau, and Li-Yau on complete manifolds with Ricci bounded from below. We also obtain an analogue of the Bonnet-Myers theorem. In Section :appendix we construct large classes of sub-Riemannian manifolds with transverse symmetries which satisfy the generalized curvature-dimension inequality . 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.