Weyl–Schouten theorem

In the mathematical field of differential geometry, the existence of isothermal coordinates for a (pseudo-)Riemannian metric is often of interest. In the case of a metric on a two-dimensional space, the existence of isothermal coordinates is unconditional. For higher-dimensional spaces, the Weyl–Schouten theorem (named after Hermann Weyl and Jan Arnoldus Schouten) characterizes the existence of isothermal coordinates by certain equations to be satisfied by the Riemann curvature tensor of the metric.

Existence of isothermal coordinates is also called conformal flatness, although some authors refer to it instead as local conformal flatness; for those authors, conformal flatness refers to a more restrictive condition.