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

Hubert Gollek — 2005

Banach Center Publications

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...

Natural algebraic representation formulas for curves in ℂ³

Hubert Gollek — 2002

Banach Center Publications

We consider several explicit examples of solutions of the differential equation Φ₁’²(z) + Φ₂’²(z) + Φ₃’²(z) = d²(z) of meromorphic curves in ℂ³ with preset infinitesimal arclength function d(z) by nonlinear differential operators of the form (f,h,d) → V(f,h,d), V = (Φ₁,Φ₂,Φ₃), whose arguments are triples consisting of a meromorphic function f, a meromorphic vector field h, and a meromorphic differential 1-form d on an open set U ⊂ ℂ or, more general, on a Riemann surface Σ. Most of them are natural...