Geometry of scrolls
Geometry of scrolls
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卷轴的几何形状
DOI：
10.18910/6363
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发表时间：
1996
影响因子：
0.4
通讯作者：
M. Umehara
中科院分区：
文献类型：
--
作者：
Osamu Kobayashi;M. Umehara
本文研究了平面R（或球面jRu{αo}）中浸入曲线的拓扑和几何.从拓扑的观点出发，我们用几何拓扑来区分曲线（[6]，[2]）。如果两条曲线的邻域之间存在一个带一个到另一个的同构，则称这两条曲线是拓扑的。地理副本保留交点信息。如果我们把自己限制在正常曲线（[8]），即，对于自交点为横截双点的曲线，交点信息用高斯字表示。一个高斯词仅仅是一个符号交叉点的标签序列。相反，高斯字决定曲线的几何类型。在几何方面，我们看曲线的顶点。顶点是曲率的不动点。众所周知，顶点是一个属于莫比乌斯几何的概念。也就是说，它不仅在欧几里得运动下不变，而且在反演下不变。我们假设曲线只有1000个顶点，没有一个顶点位于交叉点（参见。定理2.5）。然后将曲线划分为有限多个无顶点曲线。由于平面上的无顶点曲线没有自相交（Kneser，参见[5]），原始曲线的拓扑复杂性来自这些无顶点段的相交。作为一个基本的情况下，我们研究两个顶点自由曲线的交点。
In this paper, we study topology and geometry of immersed curves in the plane R (or preferably in the sphere jRu{αo}). From topological point of view, we distinguish curves by geotopy ([6], [2]). Two curves are said to be geotopic if there is a diffeomorphism between neighborhoods of the curves which takes one to the other. Geotopy preserves information on intersections. If we restrict ourselves to normal curves ([8]), i.e., curves whose self-intersections are transvers double points, the intersection information is represented by a Gauss word. A Gauss word is simply a sequence of labels of crossing points with signs. Conversely, a Gauss word determines a geotopy type of curves. On the geometric side, we look at vertices of a curve. A vertex is a stationary point of the curvature. It is well-known that a vertex is a concept which belongs to Mobius geometry. That is, it is invariant not only under Euclidean motions, but also under inversions. We assume that curves have only finitely many vertices, none of which are located at crossings (cf. Theorem 2.5). Then a curve is divided into finitely many vertex-free curves. Since a vertex-free curve on the plane has no self-intersections (Kneser, see[5]), topological complexity of the original curve then comes from intersections of these vertex-free pieces. As a basic case, we investigate intersections of two vertex-free curves.