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Concept
Isothermal coordinates
Summary
In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form
where is a positive smooth function. (If the Riemannian manifold is oriented, some authors insist that a coordinate system must agree with that orientation to be isothermal.)
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.
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. The construction used by Gauss made use of the Cauchy–Kowalevski theorem, so that his method is fundamentally restricted to the real-analytic context. Following innovations in the theory of two-dimensional partial differential equations by Arthur Korn, Leon Lichtenstein found in 1916 the general existence of isothermal coordinates for Riemannian metrics of lower regularity, including smooth metrics and even Hölder continuous metrics.
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. The angle-preserving property together with orientation-preservation is one characterization (among many) of holomorphic functions, and so an oriented coordinate atlas consisting of isothermal coordinate charts may be viewed as a holomorphic coordinate atlas.
Official source
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