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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.

kevin
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    That can always be done. Indeed you can also have $E=G$. It's called the isothermal coordinate. see http://en.wikipedia.org/wiki/Isothermal_coordinates –  Apr 13 '14 at 22:43
  • Thanks! I have to learn what Weyl and Cotton tensors are, but that did answer my question! – kevin Apr 13 '14 at 22:52

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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)

kevin
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