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 [133]. 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