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

We deal here with the geometry of the so-called twistor fibration Z→S13 over the De Sitter 3-space, where the total space ZZ is a five-dimensional reductive homogeneous space with two canonical invariant almost CR structures. Fixed the normal metric on ZZ we study the harmonic map equation for smooth maps of Riemann surfaces into ZZ. A characterization of spacelike surfaces with harmonic twistor lifts to ZZ is obtained. Also it is shown that the harmonic map equation for twistor lifts can be formulated as the curvature vanishing of an S1S1-loop of connections i.e. harmonic twistor lifts exist within S1S1-families. Special harmonic maps such as holomorphic twistor lifts are also considered and some remarks concerning (compact) vacua of the twistor energy are given.