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

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 $d\ge 1$.

arXiv comment34 pages, v2: fixed and clarified the proof of Theorem 7 and some typos, v3: final version, to appear on Journal of the Institute of Mathematics of Jussieu

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Cited by in corpus (8)

• Sub-Riemannian curvature in contact geometry
• On Jacobi fields and canonical connection in sub-Riemannian geometry
• Bakry-Émery curvature and model spaces in sub-Riemannian geometry
• H-type foliations
• Affine connections and curvature in sub-Riemannian geometry
• Comparison theorems on H-type sub-Riemannian manifolds
• Generalized Bakry-Émery curvature condition and equivalent entropic inequalities in groups
• Unified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures