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.

The Frobenius theorem[4] states that the above equation is locally solvable if and only if is symmetric in i and k for any 1-form ω.

As such, under the given curvature and dimension conditions, there always exists a locally defined 1-form ω solving the given equation.