Conformal geometry of isotropic curves in the complex quadric

Abstract

Let [Formula: see text] be the complex 3-quadric endowed with its standard complex conformal structure. We study the complex conformal geometry of isotropic curves in [Formula: see text]. By an isotropic curve, we mean a nonconstant holomorphic map from a Riemann surface into [Formula: see text], null with respect to the conformal structure of [Formula: see text]. The relations between isotropic curves and a number of relevant classes of surfaces in Riemannian and Lorentzian spaceforms are discussed.