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

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

2009-10-29

arxiv.org/abs/0910.5626v2

Mathematics

Differential Geometry

25 pages, this version v2, reference Akutagawa added, some typos and short omissions corrected. Final version to appear in Dif

Scientific paper

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.

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