Sub-Riemannian curvature in contact geometry

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 obtain a sub-Riemannian version of the Bonnet-Myers theorem that applies to any contact manifold.

31 pages, 2 figures; v2: the Bonnet-Myers theorem 1.7 now holds for any contact structure; v3: final version (with expanded introduction) to appear on Journal of Geometric Analysis; v4: fixed typos

References in corpus (3)

• On Jacobi fields and canonical connection in sub-Riemannian geometry
• Measure contraction properties of Carnot groups
• Sub-Riemannian Ricci curvatures and universal diameter bounds for 3-Sasakian manifolds

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