An elementary proof of Small's formula for null curves in PSL(2,C) and an analogue for Legendrian curves in PSL(2,C)
An elementary proof of Small's formula for null curves in PSL(2,C) and an analogue for Legendrian curves in PSL(2,C)
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发表时间：
2002-09
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通讯作者：
M. Kokubu;M. Umehara;Kotaro Yamada
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作者：
M. Kokubu;M. Umehara;Kotaro Yamada
对于PSL（2，C）中的零曲线，存在一个用两个亚纯函数及其导数表示的公式（Small公式）。本文给出了Small公式的一个初等证明。此外，还给出了PSL（2，C）中Legendrian曲线的一个类似公式.由于PSL（2，C）中的零曲线与双曲三维空间H^3中的平均曲率为1的曲面有关，勒让德曲线与H^3中的平坦曲面有关。因此，作为Lengendrian曲线的Small-type公式的应用，我们给出了H^3中平坦曲面的新例子。
For null curves in PSL(2,C), there exists a representation formula in terms of two meromorphic functions and their derivatives (Small's formula). In this paper, we give an elementary proof of Small's formula. Moreover, a similar formula for Legendrian curves in PSL(2,C) is given. As null curves in PSL(2,C) are related to mean curvature one surfaces in hyperbolic 3-space H^3, Legendrian curves are related to flat surfaces in H^3. So, as an application of Small-type formula for Lengendrian curves, we give new examples of flat surfaces in H^3.