Calculus of Variations and Geometric Measure Theory
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L. Rizzi - P. Silveira

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

created by rizzi1 on 28 Sep 2015
modified on 28 Oct 2015

[BibTeX]

Submitted Paper

Inserted: 28 sep 2015
Last Updated: 28 oct 2015

Pages: 33
Year: 2015
Notes:

v2: fixed and clarified the proof of Theorem 7 and some typos


Links: arXiv preprint

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


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