An Inverse Problem from Sub-Riemannian Geometry

Details An Inverse Problem from Sub-Riemannian Geometry An Inverse Problem from Sub-Riemannian Geometry

2001-04-14

arxiv.org/abs/math/0104157v1

Mathematics

Differential Geometry

13 pages

Scientific paper

The geodesics for a sub-Riemannian metric on a three-dimensional contact manifold $M$ form a 1-parameter family of curves along each contact direction. However, a collection of such contact curves on $M$, locally equivalent to the solutions of a fourth-order ODE, are the geodesics of a sub-Riemannian metric only if a sequence of invariants vanish. The first of these, which was earlier identified by Fels, determines if the differential equation is variational. The next two determine if there is a well-defined metric on $M$ and if the given paths are its geodesics.

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