  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.
