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, with 1$]]>, has sub-Riemannian diameter bounded by. When, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure on of the quaternionic Hopf fibrations: whose exact sub-Riemannian diameter is π, for all d ≤ 1.

SUB-RIEMANNIAN RICCI CURVATURES and UNIVERSAL DIAMETER BOUNDS for 3-SASAKIAN MANIFOLDS / Rizzi, Luca; Silveira Diaz, Pavel. - In: JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU. - ISSN 1474-7480. - 18:4(2019), pp. 783-827. [10.1017/S1474748017000226]

SUB-RIEMANNIAN RICCI CURVATURES and UNIVERSAL DIAMETER BOUNDS for 3-SASAKIAN MANIFOLDS

Luca Rizzi;Pavel Silveira
2019

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, with 1$]]>, has sub-Riemannian diameter bounded by. When, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure on of the quaternionic Hopf fibrations: whose exact sub-Riemannian diameter is π, for all d ≤ 1.
18
4
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827
http://arxiv.org/abs/1509.05410v3
Rizzi, Luca; Silveira Diaz, Pavel
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/128686
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