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

• Autorzy: Hubert Gollek
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
• 68U05: Computer graphics; computational geometry
• 53A10: Minimal surfaces, surfaces with prescribed mean curvature

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2005
• Tom: 69
• Numer: 1
• Strony: 221-242

Daty

wydano

2005

Twórcy

autor

Hubert Gollek

• Institute of Mathematics, Humboldt University, Sitz: Rudower Chaussee 25, 10099 Berlin, Germany

Identyfikatory

DOI

10.4064/bc69-0-18