论文标题
表面上的离散线字段
Discrete Line Fields on Surfaces
论文作者
论文摘要
向量字段和线场，它们的对应物在切线线上没有方向，是动态系统理论中熟悉的对象。在他们的研究中使用的技术中，（通用）字段的摩尔斯 - 摩尔分解起着基本作用，将相空间的几何结构与由关键点和分离的组合对象相关联。这些概念使福尔曼（Forman）与连续情况相比，使离散媒介领域的理论令人满意。在本文中，我们引入了离散的线场。同样，我们的定义足够丰富，可以在连续线场理论中提供基本结果的对应物：Euler-Poincaré公式，莫尔斯 - 摩尔分解和对关键元素的拓扑一致的取消，从而允许对原始离散线路的拓扑简化。
Vector fields and line fields, their counterparts without orientations on tangent lines, are familiar objects in the theory of dynamical systems. Among the techniques used in their study, the Morse--Smale decomposition of a (generic) field plays a fundamental role, relating the geometric structure of phase space to a combinatorial object consisting of critical points and separatrices. Such concepts led Forman to a satisfactory theory of discrete vector fields, in close analogy to the continuous case. In this paper, we introduce discrete line fields. Again, our definition is rich enough to provide the counterparts of the basic results in the theory of continuous line fields: a Euler-Poincaré formula, a Morse--Smale decomposition and a topologically consistent cancellation of critical elements, which allows for topological simplification of the original discrete line field.
