Conformal geometry of isotropic curves in the complex quadric

Abstract

Let ${\mathbb{Q}}_3$ be the complex 3-quadric endowed with its standard complex conformal structure. We study the complex conformal geometry of isotropic curves in ${\mathbb{Q}}_3$. By an isotropic curve we mean a nonconstant holomorphic map from a Riemann surface into ${\mathbb{Q}}_3$, null with respect to the conformal structure of ${\mathbb{Q}}_3$. The relations between isotropic curves and a number of relevant classes of surfaces in Riemannian and Lorentzian spaceforms are discussed.

26 pages, 5 figures