# Algebraic representation formulas for null curves in Sl(2,ℂ)

Banach Center Publications (2005)

- Volume: 69, Issue: 1, page 221-242
- ISSN: 0137-6934

## Access Full Article

top## Abstract

top## How to cite

topHubert Gollek. "Algebraic representation formulas for null curves in Sl(2,ℂ)." Banach Center Publications 69.1 (2005): 221-242. <http://eudml.org/doc/282170>.

@article{HubertGollek2005,

abstract = {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.},

author = {Hubert Gollek},

journal = {Banach Center Publications},

keywords = {Bianchi-Small formula; representation formulas; constant mean curvature 1},

language = {eng},

number = {1},

pages = {221-242},

title = {Algebraic representation formulas for null curves in Sl(2,ℂ)},

url = {http://eudml.org/doc/282170},

volume = {69},

year = {2005},

}

TY - JOUR

AU - Hubert Gollek

TI - Algebraic representation formulas for null curves in Sl(2,ℂ)

JO - Banach Center Publications

PY - 2005

VL - 69

IS - 1

SP - 221

EP - 242

AB - 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.

LA - eng

KW - Bianchi-Small formula; representation formulas; constant mean curvature 1

UR - http://eudml.org/doc/282170

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.