曲面上的离散线场
Discrete Line Fields on Surfaces
摘要：
向量场和线场是动力系统理论中常见的对象，它们在切线上没有方向。在他们的研究中使用的技术中，Morse-Smale分解的（通用）字段起着根本性的作用，相空间的几何结构与由临界点和分界线组成的组合对象有关。这些概念导致福尔曼一个令人满意的理论离散向量场，在密切类似的连续情况下。在本文中，我们引入离散线场。同样，我们的定义是丰富的，足以提供对应的基本结果在理论的连续线场：一个欧拉-庞加莱公式，Morse-Smale分解和拓扑一致的取消的关键元素，它允许拓扑简化的原始离散线场。
英文摘要：
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.
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