This means that in isothermal coordinates, the Riemannian metric locally has the form where
Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold.
By contrast, most higher-dimensional manifolds do not admit isothermal coordinates anywhere; that is, they are not usually locally conformally flat.
In dimension 3, a Riemannian metric is locally conformally flat if and only if its Cotton tensor vanishes.
In dimensions > 3, a metric is locally conformally flat if and only if its Weyl tensor vanishes.
In 1822, Carl Friedrich Gauss proved the existence of isothermal coordinates on an arbitrary surface with a real-analytic Riemannian metric, following earlier results of Joseph Lagrange in the special case of surfaces of revolution.
[1] The construction used by Gauss made use of the Cauchy–Kowalevski theorem, so that his method is fundamentally restricted to the real-analytic context.
[3] Given a Riemannian metric on a two-dimensional manifold, the transition function between isothermal coordinate charts, which is a map between open subsets of R2, is necessarily angle-preserving.
Furthermore, given an oriented surface, two Riemannian metrics induce the same holomorphic atlas if and only if they are conformal to one another.
By the 1950s, expositions of the ideas of Korn and Lichtenstein were put into the language of complex derivatives and the Beltrami equation by Lipman Bers and Shiing-shen Chern, among others.
[4] In this context, it is natural to investigate the existence of generalized solutions, which satisfy the relevant partial differential equations but are no longer interpretable as coordinate charts in the usual way.
This was initiated by Charles Morrey in his seminal 1938 article on the theory of elliptic partial differential equations on two-dimensional domains, leading later to the measurable Riemann mapping theorem of Lars Ahlfors and Bers.
[7][8] A simpler approach to the Beltrami equation has been given more recently by Adrien Douady.
satisfies so that the coordinates (u, v) will be isothermal if the Beltrami equation has a diffeomorphic solution.
The existence of isothermal coordinates on a smooth two-dimensional Riemannian manifold is a corollary of the standard local solvability result in the analysis of elliptic partial differential equations.
In the present context, the relevant elliptic equation is the condition for a function to be harmonic relative to the Riemannian metric.
The local solvability then states that any point p has a neighborhood U on which there is a harmonic function u with nowhere-vanishing derivative.