Let $(M^2,g)$ be a 2-dimensional Riemannian manifold. For any point $p\in M^2$ can we always find coordinates $(u,v)$ in a neighborhood $U$ of $p$ such that the Gaussian curvature is only a function of $u\pm v$, i.e, $K=K(u\pm v)$ ?



Can we always find coordinates on a surface such that $K=K(u-v)$?
Curvature is a smooth function on the surface, and locally, any smooth negative function can serve as a curvature of some surface (M. S. Berger, Riemannian structure of prescribed Gaussian curvature for compact 2-manifolds, J. Differential Geom. 5 (1971), 325-332.)


So the question is whether for an arbitrary smooth function there exists
a coordinate system such that the function depends on only one coordinate. The answer is certainly negative, since the function can have critical points.


