标题： 从子黎曼几何中的反问题 显示英文标题

标题： An Inverse Problem from Sub-Riemannian Geometry

摘要： 三维接触流形上的子黎曼度量的测地线$M$沿每个接触方向形成一个1参数曲线族。 然而，$M$上的一组这样的接触曲线，局部等价于四阶常微分方程的解，只有当一系列不变量消失时，才是子黎曼度量的测地线。 其中第一个不变量，先前由Fels确定，用于判断微分方程是否为变分的。 接下来的两个不变量用于判断在$M$上是否存在定义良好的度量以及给定路径是否为其测地线。 显示英文摘要

摘要： 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.