Connections and Curvature in sub-Riemannian geometry

Details Connections and Curvature in sub-Riemannian geometry Connections and Curvature in sub-Riemannian geometry

2009-12-17

arxiv.org/abs/0912.3535v2

Mathematics

Differential Geometry

One section removed. Two new sections added

Scientific paper

For a subRiemannian manifold and a given Riemannian extension 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 manifolds to the subRiemannian 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 subRiemannian generalizations of the Bonnet-Myers theorem, providing some new results and some new proofs and interpretations of existing results.

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