SUB-RIEMANNIAN RICCI CURVATURES and UNIVERSAL DIAMETER BOUNDS for 3-SASAKIAN MANIFOLDS

For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev-Zelenko-Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet-Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider the sub-Riemannian structure of 3-Sasakian manifolds, for which we provide explicit curvature formulas. We prove that any complete 3-Sasakian structure of dimension, with 1$]]>, has sub-Riemannian diameter bounded by. When, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure on of the quaternionic Hopf fibrations: whose exact sub-Riemannian diameter is π, for all d ≤ 1.