CONNECTIONS AND CURVATURE IN SUB-RIEMANNIAN GEOMETRY

Abstract. For a sub-Riemannian manifold and a given Riemannian exten-sion of the metric, we define a canonical global connection. This connection coincides with both the Levi-Civita connection on Riemannian manifolds and the Tanaka-Webster connection on strictly pseudoconvex CR manifolds. We define a notion of normality generalizing Tanaka’s notion for CR man-ifolds to the sub-Riemannian case. Under the assumption of normality, we construct local frames that simplify computations in a manner analogous to Riemannian normal coordinates. We then use these frames to establish Bianchi Identities and symmetries for the associated curvatures. Finally we explore sub-Riemannian generalizations of the Bonnet-Myers theorem, pro-viding some new results and some new proofs and interpretations of existing results. 1