Duals of Vector Fields and of Null Curves

Abstract

The notion of dual curve of projective differential geometry is generalized to a dualization operator on the space of curves in the tangent bundle of a real or complex Riemannian manifold M and on the space of vector fields too. In dimension 3 it turns out that dualization maps null curves to isotropic curves and vice versa. We study this dualization in the special cases of $${M = \mathbb{C}^3}$$ and $${M = \rm{Sl}(2, \mathbb{C})}$$ and give analytic descriptions in terms of the Weierstras-, respectively Bryant data, of null curves in $${M = \mathbb{C}^3}$$ resp. $${M = \rm{Sl}(2, \mathbb{C})}$$ .