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Let be a smooth connected manifold endowed with a smooth measure and a smooth locally subelliptic diffusion operator satisfying , and which is symmetric with respect to . Associated with one has the carré du champ and a canonical distance , with respect to which we suppose that be complete. We assume that is also equipped with another first-order differential bilinear form and we assume that and satisfy the Hypothesis 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.
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Fabrice Baudoin, Nicola Garofalo, Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries. J. Eur. Math. Soc. 19 (2017), no. 1, pp. 151–219DOI 10.4171/JEMS/663