  In the paper we consider Riemannian surfaces admitting a global expression of
the Gauss curvature as the divergence of a vector field. It is equivalent to
the existence of a metric linear connection of zero curvature. Such a linear
connection $\nabla$ plays an important role in the differential geometry of
non-Riemannian surfaces in the sense that the Riemannian quadratic forms can be
changed into Minkowski functionals in the tangent planes such that the
Minkowskian length of the tangent vectors is invariant under the parallel
translation with respect to $\nabla$ (compatibility condition). A smoothly
varying family of Minkowski functionals in the tangent planes is called a
Finslerian metric function under some regularity conditions. Especially, the
existence of a compatible linear connection provides the Finsler surface to be
a so-called generalized Berwald surface. It is an alternative of the Riemannian
geometry for $\nabla$. Using some general observations and topological
obstructions we concentrate on explicit examples. In some representative cases
(Euclidean plane, hyperbolic plane etc.) we solve the differential equation of
the parallel vector fields to construct a smoothly varying family of Minkowski
functionals in the tangent planes such that the Minkowskian length of the
tangent vectors is invariant under the parallel translation.
