Sub-Riemannian Ricci curvatures and universal diameter bounds for 3-Sasakian manifolds

TL;DR

This paper introduces canonical Ricci curvatures for fat sub-Riemannian structures, proves comparison theorems and diameter bounds for 3-Sasakian manifolds, and demonstrates the sharpness of these bounds on quaternionic Hopf fibrations.

Contribution

It defines new Ricci curvature invariants for fat sub-Riemannian structures and establishes sharp diameter bounds for 3-Sasakian manifolds, extending classical Riemannian results.

Findings

Sub-Riemannian diameter of certain 3-Sasakian manifolds is bounded by π.

Explicit curvature formulas are provided for 3-Sasakian structures.

Results are sharp for quaternionic Hopf fibrations.

Abstract

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{HP}^d, \end{equation*} whose exact sub-Riemannian diameter is $\pi$, for all $d \geq 1$.