Orthogonal Parametrization of a Regular Surface

Question

I was just wondering whether or not it is always possible to parametrize a regular surface $S$ via a function $X$ of local coordinates $u$, $v$ such that $X$ is an orthogonal parametrization- that is to say, such that the first fundamental form has $F=0$ everywhere ($\forall (u,v) \in R$, where the domain of $X$ is $R$). If so, prove it. If not, give a counterexample and explain the most general case of a surface (known to you) of when it is possible.

Answer 1

Isothermal coordinates on surfaces were first introduced by Gauss. Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold. On higher-dimensional Riemannian manifolds a necessary and sufficient condition for their local existence is the vanishing of the Weyl tensor and of the Cotton tensor. (From Wikipedia: http://en.wikipedia.org/wiki/Isothermal_coordinates)