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 $4d+3$, with $d>1$, has sub-Riemannian diameter bounded by $\pi$. When $d=1$, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure on ${\mathbb{S}}^{4d+3}$ of the quaternionic Hopf fibrations: \begin{equation*} $\mathbb{S}$^3 \hookrightarrow $\mathbb{S}$^{4d+3} \to $\mathbb{H}\mathbb{P}$^d, \end{equation*} whose exact sub-Riemannian diameter is $\pi$, for all $d\ge 1$.