Published Paper

Inserted: 28 sep 2015
Last Updated: 17 aug 2018

Journal: Journal of the Institute of Mathematics of Jussieu
Pages: 34
Year: 2015
Doi: 10.1017/S1474748017000226

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: \[ \mathbb{S}^3 \hookrightarrow \mathbb{S}^{4d+3} \to \mathbb{HP}^d, \] whose exact sub-Riemannian diameter is $\pi$, for all $d \geq 1$.

Keywords: sub-Riemannian, sub-Laplacian, hypoelliptic, comparison, Sasakian, Bonnet-Myers