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 subRiemannian 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.

Volume and distance comparison theorems for sub-Riemannian manifolds

“…The latter, in particular, recovers and extends similar results obtained in the recent literature, with an unified and simpler proof. We refer in particular to [2,Thm. 1.7], for contact structures [6, Thm.…”

A canonical connection on sub-Riemannian contact manifolds

“…In particular, Agrachev, Barilari, and Rizzi [1] lament the lack of a canonical connection 'à la Levi-Civita,' instead deriving curvature invariants by other means, especially via Jacobi fields. These authors are working on contact manifolds but the approach via Jacobi fields is quite general, as shown by Barilari and Rizzi [3], following earlier work of Zelenko and Li [12].…”

“…In Riemannian geometry, perturbations of a given geodesic γ are controlled by Jacobi fields J a satisfying the Jacobi equation so the curvature tensor may, in some sense, be read off from the Jacobi equation, better so from the perturbation of extremals up on T * M (more usually called the geodesic spray). In the sub-Riemannian case, the upshot is that 'curvature invariants' may be similarly read off from a 'Jacobi equation' on T * M as in [1,3,12]. In [3] it is shown that these curvature invariants may also be derived from the (necessarily non-linear) Ehresmann connection introduced in [12].…”

“…We should like to thank the referee for drawing our attention to [1,3,12] and for suggesting various corrections.…”

Bakry–Émery curvature and model spaces in sub-Riemannian geometry

“…Sasakian manifolds. We follow the notation of [ABR17], to which we refer to for details and references. A contact manifold (M, ω) is a smooth odd-dimensional manifold endowed with a 1-form such that dω is non-degenerate on ker ω.…”