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SUB-RIEMANNIAN RICCI CURVATURES AND UNIVERSAL DIAMETER BOUNDS FOR 3-SASAKIAN MANIFOLDS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 18 / Issue 4 / July 2019
- Published online by Cambridge University Press:
- 21 June 2017, pp. 783-827
- Print publication:
- July 2019
-
- Article
- Export citation
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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 
$\unicode[STIX]{x1D70B}$. 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: whose exact sub-Riemannian diameter is
$$\begin{eqnarray}\mathbb{S}^{3}{\hookrightarrow}\mathbb{S}^{4d+3}\rightarrow \mathbb{HP}^{d},\end{eqnarray}$$
$\unicode[STIX]{x1D70B}$, for all 
$d\geqslant 1$.
