Self-Crossing Geodesics
Self-Crossing Geodesics
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自相交测地线
DOI：
10.1007/s00283-021-10127-0
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
Petrunin, Anton
中科院分区：
文献类型：
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作者：
Petrunin, Anton
将一根橡皮筋拉伸在卵石上，并在多个点上交叉（见图 1）。自交叉的组合可以用一条闭合平面曲线来描述——它是表面参数化中删除一个点的橡皮筋。例如，如果你可以转动卵石，你会看到自交叉是由图 1 (b) 中的平面曲线描述的。我们假设卵石的表面是强凸的、光滑的且无摩擦的；在本例中，橡皮筋模拟了一条闭合测地线。假设我们对自我交叉的可能模式感兴趣。更准确地说，强凸光滑闭合曲面上的闭合测地线自交叉的可能组合类型是什么？
A rubber band is stretched around a pebble, and it crosses itself at several points (see Figure 1). The combinatorics of self-crossings can be described by a closed plane curve—it is the rubber band in a parameterization of the surface with one point removed. For example, if you could turn the pebble around, you would see that the self-crossings are described by the plane curve in Figure 1 (b).We assume that the surface of the pebble is strongly convex, smooth, and frictionless; in this case, the rubber band models a closed geodesic. Suppose that we are interested in possible patterns of self-crossings. More precisely, what are the possible combinatoric types of selfcrossings of a closed geodesic on a strongly convex smooth closed surface?