We compare different notions of curvature on contact sub-Riemannian manifolds. In particular we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the sub-Riemannian distance. We explicitly compute their expressions in terms of the standard tensors of contact geometry. As an application of these results, we prove a version of the sub-Riemannian Bonnet-Myers theorem that applies to any contact manifold, with special attention to contact Yang-Mills structures.
Sub-Riemannian curvature in contact geometry / Agrachev, Andrey; Barilari, Davide; Rizzi, Luca. - In: THE JOURNAL OF GEOMETRIC ANALYSIS. - ISSN 1050-6926. - 27:1(2017), pp. 366-408. [10.1007/s12220-016-9684-0]
Sub-Riemannian curvature in contact geometry
Agrachev, Andrey;Rizzi, Luca
2017-01-01
Abstract
We compare different notions of curvature on contact sub-Riemannian manifolds. In particular we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the sub-Riemannian distance. We explicitly compute their expressions in terms of the standard tensors of contact geometry. As an application of these results, we prove a version of the sub-Riemannian Bonnet-Myers theorem that applies to any contact manifold, with special attention to contact Yang-Mills structures.File | Dimensione | Formato | |
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