The Geometry of Sub-riemannian Three-manifolds
نویسنده
چکیده
The local equivalence problem for sub-Riemannian structures on threemanifolds is solved. In the course of the solution, it is shown how to attach a canonical Riemannian metric and connection to the given sub-Riemannian metric and it is shown how all of the differential invariants of the sub-Riemannian structure can be calculated. The relation between the completeness of the sub-Riemannian metric, the associated Riemannian metric, and geodesic completeness is investigated, and an example is given of a manifold that is complete in the sub-Riemannian metric but not complete in the canonical associated Riemannian metric. It is shown that the Jacobi equations for subRiemannian geodesics can be interpreted as a scalar, fourth-order, self-adjoint linear operator along each geodesic. The influence of the differential invariants of the subRiemannian structure on the conjugate points is investigated, and the results are used to prove a Bonnet-Myers-type theorem for complete sub-Riemannian 3-manifolds.
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تاریخ انتشار 2003