An elementary proof of Small's formula for null curves in PSL(2,C) and an analogue for Legendrian curves in PSL(2,C)

By M. Kokubu, Masaaki Umehara, Kotaro Yamada. 2002

52 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4211cc7f5fc2622f60738f9405c3dc16d2982501

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Kentaro Saji
2. Masaaki Umehara
3. Kotaro Yamada

In Geometry and Topology

1. Kentaro Saji
2. Masaaki Umehara
3. Kotaro Yamada

Citations in the corpus

• The geometry of fronts (2005, methodology): “Recently, global properties of fronts were investigated via flat surfaces in hyperbolic 3-space H3 ([ KUY1 , KRSUY]), via maximal surfaces in Minkowski 3-space”