Abstract
We deal here with the geometry of the so-called twistor fibration Z → S 1 3 over the De Sitter 3-space, where the total space Z is a five-dimensional reductive homogeneous space with two canonical invariant almost CR structures. Fixed the normal metric on Z we study the harmonic map equation for smooth maps of Riemann surfaces into Z . A characterization of spacelike surfaces with harmonic twistor lifts to Z is obtained. Also it is shown that the harmonic map equation for twistor lifts can be formulated as the curvature vanishing of an S 1 -loop of connections i.e. harmonic twistor lifts exist within 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.
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