On the divergence representation of the Gauss curvature of Riemannian surfaces and its applications

Abstract

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.