Spacelike surfaces in De Ditter 3-space and their twistor lifts

TL;DR

This paper explores the geometry of twistor fibrations over De Sitter 3-space, characterizing spacelike surfaces with harmonic lifts and relating harmonic map equations to curvature conditions within loop families.

Contribution

It provides a new characterization of spacelike surfaces with harmonic twistor lifts and links the harmonic map equation to curvature vanishing in loop families.

Findings

Harmonic twistor lifts correspond to curvature-free connections.

Holomorphic twistor lifts are a special class of harmonic maps.

Harmonic map equations can be expressed via loop of connections.

Abstract

We deal here with the geometry of the twistor fibration $\mathcal{Z} \to \bb{S}^3_1$ over the De Sitter 3-space. The total space $\mathcal{Z}$ is a five dimensional reductive homogeneous space with two canonical invariant almost CR structures. Fixed the normal metric on $\mathcal{Z}$ we study the harmonic map equation for smooth maps of Riemann surfaces into $\mathcal{Z}$. A characterization of spacelike surfaces with harmonic twistor lifts to $\mathcal{Z}$ is obtained. It is also shown that the harmonic map equation for twistor lifts can be formulated as the curvature vanishing of an $\bb{S}^1$-loop of connections i.e. harmonic twistor lifts exist within $\bb{S}^1$-families. Special harmonic maps such as holomorphic twistor lifts are also considered and some remarks concerning (compact) vacua of the twistor energy are given.