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

Hubert 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 -