  We study curvature dimension inequalities for the sub-Laplacian on contact
Riemannian manifolds. This new curvature dimension condition is then used to
obtain: 1) Geometric conditions ensuring the compactness of the underlying
manifold (Bonnet-Myers type results); 2) Volume estimates of metric balls; 3)
Gradient bounds and stochastic completeness for the heat semigroup generated by
the sub-Laplacian; 4) Spectral gap estimates.
