标量曲率喷流的全局可度量化性
Global Metrizability on Sprays of Scalar Curvature
AI总结：
本文研究标量曲率喷流的全局可度量化性，证明在平凡第一de Rham上同调群条件下，正则标量曲率喷流全局可度量化等价于局部可度量化，并构造了圆柱面上局部可度量化但非全局可度量化 的奇异Berwald喷流例子。
AI中文摘要：
标量曲率喷流构成了一类重要的喷流，此类喷流包含所有二维喷流。本文在一定的曲率和拓扑条件下，考虑流形上某些喷流的全局可度量化性。我们证明，对于定义在维数$n\ge 3$的流形$M$上、具有几乎处处非零Ricci曲率的正则标量曲率喷流${\bf G}$，若$M$具有平凡的第一de Rham上同调群，则${\bf G}$是全局可度量化当且仅当它是局部可度量化。进一步，我们刻画了具有平凡第一de Rham上同调群的流形上一类二维奇异Berwald喷流的全局可度量化性。同时，在圆柱面$S^1\times {\bf R}$（具有非平凡的第一de Rham上同调群）上，我们构造了一族局部可度量化但非全局可度量化 的奇异Berwald喷流。最后，我们在圆柱面或球面上构造了一些具有特殊性质的二维喷流的例子。
英文摘要：
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.
