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 π. When d = 1, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure of the quaternionic Hopf fibrations on the 4d+3 dimensional sphere, whose exact sub-Riemannian diameter is π, for all d ≥ 1.
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Submitted on : Wednesday, October 28, 2015 - 1:27:25 PM
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Luca Rizzi, Pavel Silveira. Sub-Riemannian Ricci curvatures and universal diameter bounds for 3-Sasakian manifolds. Journal of the Institute of Mathematics of Jussieu, Cambridge University Press (CUP), In press, ⟨10.1017/S1474748017000226⟩. ⟨hal-01221661⟩

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