We study curves in Sl(2,ℂ) whose tangent vectors have vanishing length with respect to the biinvariant conformal metric induced by the Killing form, so-called null curves. We establish differential invariants of them that resemble infinitesimal arc length, curvature and torsion of ordinary curves in Euclidean 3-space. We discuss various differential-algebraic representation formulas for null curves. One of them, a modification of the Bianchi-Small formula, gives an Sl(2,ℂ)-equivariant bijection between pairs of meromorphic functions and null curves. The inverse of this formula is also differential-algebraic. The other one is based on an integral formula deduced from that of R. Bryant, using certain natural differential operators on Riemannian surfaces that we introduced in [7] for differential-algebraic representation formulas of curves in ℂ³. We demonstrate some commands of a Mathematica package that resulted from our investigations, containing algebraic and graphical utilities to handle null curves, their invariants, representation formulas and associated surfaces of constant mean curvature 1 in ℍ³, taking into consideration several models of ℍ³.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc69-0-18,
author = {Hubert Gollek},
title = {Algebraic representation formulas for null curves in Sl(2,$\mathbb{C}$)},
journal = {Banach Center Publications},
volume = {68},
year = {2005},
pages = {221-242},
zbl = {1144.53304},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc69-0-18}
}
Hubert Gollek. Algebraic representation formulas for null curves in Sl(2,ℂ). Banach Center Publications, Tome 68 (2005) pp. 221-242. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc69-0-18/