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

Details 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)

2002-09-20

arxiv.org/abs/math/0209258v2

Mathematics

Differential Geometry

13 pages, 9 figures

Scientific paper

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.

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