具有切向边界条件的凸多面体中单位矢量场的分类
Classification of unit-vector fields in convex polyhedra with tangent boundary conditions
摘要：
一个三维凸多面体P上的单位向量场n是相切的，如果在P的面上，n与这些面相切。给出了离P的顶点连续的切单位向量场的同伦分类。分类由某些不变量确定，即边缘方向（P的边缘上的n的值），扭结数（P的面上的边缘之间的n的相对缠绕数）和缠绕数（P的顶点之间的表面上的n的相对度数），它们服从某些求和规则。另一个不变量，捕获面积，是用这些来表示的。这项研究的一个动机来自液晶物理学;切向单位矢量场描述了液晶在某些多面体单元中的取向。
英文摘要：
A unit-vector field n on a convex three-dimensional polyhedron P is tangent if, on the faces of P, n is tangent to the faces. A homotopy classification of tangent unit-vector fields continuous away from the vertices of P is given. The classification is determined by certain invariants, namely edge orientations (values of n on the edges of P), kink numbers (relative winding numbers of n between edges on the faces of P), and wrapping numbers (relative degrees of n on surfaces separating the vertices of P), which are subject to certain sum rules. Another invariant, the trapped area, is expressed in terms of these. One motivation for this study comes from liquid crystal physics; tangent unit-vector fields describe the orientation of liquid crystals in certain polyhedral cells.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
