K\"{a}hler manifolds with orthogonal coordinates

TL;DR

This paper characterizes 4-dimensional Kähler manifolds that admit real orthogonal coordinates, showing they are essentially products of two Riemann surfaces, thus extending the understanding of coordinate systems in complex geometry.

Contribution

It proves that the only 4-dimensional Kähler manifolds with real orthogonal coordinates are, up to a cover, products of two Riemann surfaces, providing a classification result.

Findings

Only 4D Kähler manifolds with orthogonal coordinates are products of Riemann surfaces.

Orthogonal coordinates simplify local expressions for differential operators and curvature.

The result extends known coordinate existence results to complex geometry contexts.

Abstract

If one could assume that local coordinates in a Riemannian manifold were orthogonal, then local expressions for differential operators, and curvature computations, would be simplified. It is always possible on 2-manifolds, using geometric normal coordinates or isothermal coordinates. In 1984, Dennis DeTurck and Dean Yang constructed smooth orthogonal coordinates on any Riemannian 3-manifold. In fact, they showed that, at any point in a 3-manifold, and for any orthonormal frame at that point, there is a set of local orthogonal coordinates so that the partial derivatives at that point are that frame. Recently, Paul Gauduchon and Andrei Moroianu showed, by contrast, that there are no orthogonal coordinates on $\mathbb{CP}^{n}$ or $\mathbb{HP}^{n}$. Their proof strongly uses the simplicity of the curvature tensor for these spaces. The main theorem of the present article is to show that the only 4 real-dimensional K\"{a}hler manifolds which admit real orthogonal coordinates are, up to a cover, a Riemannian product of 2 Riemann surfaces.