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 Weierstraß-, respectively Bryant data, of null curves in \({M = \mathbb{C}^3}\) resp. \({M = \rm{Sl}(2, \mathbb{C})}\).
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Received: March 22, 2006.
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Gollek, H. Duals of Vector Fields and of Null Curves. Result. Math. 50, 53–79 (2007). https://doi.org/10.1007/s00025-006-0235-z
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DOI: https://doi.org/10.1007/s00025-006-0235-z