Twistor theory of Loxodromes

Abstract

The twistor correspondence provides a duality between curves in $S^4$ and ruled surfaces in ${\mathbb{CP}}^3$ in which the conformal properties of the former are reflected in the projective properties of the latter. We use this to characterise a $14$--dimensional family of homogeneous ruled surfaces in ${\mathbb{CP}}^3$ which correspond to loxodromes in $S^4$.

Dedicated to Roger Penrose on the occasion of his 95th birthday