Algebraic representation formulas for null curves in ${\rm Sl}(2,{\Bbb C})$
Tom 69 / 2005
Streszczenie
We study curves in ${\rm Sl}(2,{\Bbb C})$ 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 ${\rm Sl}(2,{\Bbb C})$-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 \cite{gollek4} for differential-algebraic representation formulas of curves in ${\mathbb C}^3$. 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 ${\mathbb H}^3$, taking into consideration several models of ${\mathbb H}^3$.
