Conformal geometry of isotropic curves in the complex quadric

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