GLOBAL PROPERTIES OF SPACELIKE CURVES IN MINKOWSKI 3-SPACE
GLOBAL PROPERTIES OF SPACELIKE CURVES IN MINKOWSKI 3-SPACE
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DOI：
10.1142/s0218216506004828
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发表时间：
2006-09
影响因子：
0.5
通讯作者：
S. Izumiya;M. Kikuchi;Masatomo Takahashi
中科院分区：
文献类型：
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作者：
S. Izumiya;M. Kikuchi;Masatomo Takahashi
研究了三维Minkowski空间中闭类空曲线的整体性质。1导言最近出现了一些关于Minkowski空间中类空子流形的微分几何的文章[2，3，4，5，7，8]。一个自然的问题是，Minkowski空间中类空子流形的整体性质与欧氏空间中的子流形的性质有何不同？最简单的情况是Minkowski 3-空间中的类空曲线。本文研究了三维Minkowski空间中类空闭曲线的整体性质。在Minkowski 3-空间中存在标准欧几里得平面，其自然投影来自Minkowski 3-空间(参见第2节)。如果我们考虑这样一个平面中的一条正则(浸入)闭曲线，那么它就是一条类空间的正则闭曲线。首先，我们考虑了类空闭曲线的整体性质与闭欧氏平面曲线的整体性质有何不同的问题。对于任意正则类空曲线，其投影像都是正则平面曲线。因此，对于类空正则闭曲线，投影图像的旋转数是一个正则同伦不变量。逆断言对于正则类空闭曲线也是正确的(参见定理3.1)。根据Whitney[12]的定理，类空正则闭曲线的投影象的旋转数是关于类空正则同伦的完全不变量。我们还给出了计算类空正则闭曲线投影像旋转数的洛伦兹几何公式(参见定理3.8)。其次，我们考虑类空间结(封闭的嵌入类空间曲线)。为了避免野性，我们将其归入PL-范畴。我们认为类太空��之间的类天同构
We study global properties of closed spacelike curves in Minkowski 3-space. 1I ntroduction Recently there appeared some articles on differential geometry of spacelike submanifolds in Minkowski space [2, 3, 4, 5, 7, 8]. A natural question is how global properties of spacelike submanifolds in Minkowski space are different from those properties of submanifold in Euclidean space? The simplest case is spacelike curves in Minkowski 3-space. In this paper we study global properties of spacelike closed curves in Minkowski 3-space. There exists the canonical Euclidean plane in Minkowski 3-space with the natural projection from Minkowski 3-space (cf., §2). If we consider a regular (immersed) closed curve in such the plane, then it is a spacelike regular closed curve. Firstly we consider a problem how global properties of spacelike closed curves are different from those of closed Euclidean plane curves. For any regular spacelike curve, the projection image is a regular plane curve. Therefore the rotation number of the projection image is a regular homotopy invariant for spacelike regular closed curves. The converse assertion is also true for regular spacelike closed curves (cf., Theorem 3.1). It follows from the theorem of Whitney [12] that the rotation number of the projection image of a spacelike regular closed curve is a complete invariant with respect to the spacelike regular homotopy. We also give a Lorentzian geometric formula for calculating the rotation number of the projection image of a spacelike regular closed curve (cf., Theorem 3.8). Secondary, we consider spacelike knots (closed embedded spacelike curves). In order to avoid wildness we consider in the PL-category. We consider the spacelike isotopy among spacelike � �