Global Metrizability on Sprays of Scalar Curvature

Abstract

Sprays of scalar curvature constitute an important class of sprays, and such a class includes all two-dimensional sprays. In this paper, we consider the global metrizability of certain sprays on a manifold under some curvature and topological conditions. We prove that, for a regular spray {\bf G} of scalar curvature with almost everywhere nonzero Ricci curvature on a manifold $M$ of dimension $n\ge 3$, {\bf G} is globally metrizable if and only if it is locally metrizable, provided that $M$ has trivial first de Rham cohomology group. Further, we characterize the global metizability of a class of two-dimensional singular Berwald sprays on a manifold with trivial first de Rham cohomology group. Meanwhile, on a cylinder $S^1\times {\bf R}$ (with non-trivial first de Rham cohomology group), we construct a family of singular Berwald sprays which are locally metrizable but not globally metrizable. Finally, we construct some examples of two-dimensional sprays on cylinders or spheres with special properties.