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Let M 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 the carré du champ Γ 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 Γ Z and we assume that Γ and Γ Z satisfy the Hypothesis 1.1, 1.2, and 1.4 below. With these forms we introduce in (1.12) below a generalization of the curvaturedimension 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 WebsterTanaka-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.

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