A rigorous way to define a metric on a smooth manifold is to define a metric tensor (or Riemannian tensor), which is a quadratic function of two tangent vectors. This can be considered as an inner product on , which can be used to measure angles. This leads to the definition of the Riemannian metric, which is based on the shortest paths (called geodesics) in . An example of this appeared in the context of Lagrangian mechanics in Section 13.4.1. The kinetic energy, (13.70), serves as the required metric tensor, and the geodesics are the motions taken by the dynamical system to conserve energy. The metric can be defined as the length of the geodesic that connects a pair of points. If the chosen Riemannian metric has some physical significance, as in the case of Lagrangian mechanics, then the resulting metric provides meaningful information. Unfortunately, it may be difficult or expensive to compute its value.
Steven M LaValle 2012-04-20