论文标题
关于Riemannian表面高斯曲率的分歧表示及其应用
On the divergence representation of the Gauss curvature of Riemannian surfaces and its applications
论文作者
论文摘要
在本文中，我们将riemannian表面承认，将高斯曲率的全球表达视为矢量场的差异。它等效于存在零曲率的度量线性连接。这样的线性连接$ \ nabla $在非河流表面的差异几何形状中起着重要作用，因为可以将riemannian二次形式更改为切线平面中的minkowski功能，从而在compallate complate complate complate complate complallation complate complate to $ nabla（$ \ nabla）的情况下，相互插图的minkowskian长度是不变的。在某些规律性条件下，切线平面中的Minkowski功能的平稳族称为Finslerian度量功能。尤其是，兼容线性连接的存在使鳍表面是所谓的广义伯瓦尔德表面。它是$ \ nabla $的Riemannian几何形状的替代方法。使用一些一般的观察和拓扑障碍，我们集中于明确的例子。在某些代表性的情况下（欧几里得平面，双曲平面等），我们求解平行矢量场的微分方程，以在切线平面中构建一个平稳变化的Minkowski函数家族，以使切线矢量的Minkowskian长度在平行翻译下是不变的。
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.
