Conformal geometry of isotropic curves in the complex quadric

TL;DR

This paper investigates the complex conformal geometry of isotropic curves in the complex quadric, revealing their relationships with various classes of surfaces in different spaceforms.

Contribution

It provides a detailed study of isotropic curves in the complex quadric and explores their connections to surfaces in Riemannian and Lorentzian geometries.

Findings

Characterization of isotropic curves in the complex quadric

Relations between isotropic curves and surfaces in spaceforms

Insights into the conformal structure of 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.