Twistorial constructions of spacelike surfaces in Lorentzian 4-manifolds

Details Twistorial constructions of spacelike surfaces in Lorentzian 4-manifolds Twistorial constructions of spacelike surfaces in Lorentzian 4-manifolds

1999-02-17

arxiv.org/abs/math/9902101v1

Mathematics

Differential Geometry

Latex2.09, 26 pages

Scientific paper

We investigate the twistor space and the Grassmannian fibre bundle of a Lorentzian 4-space with natural almost optical structures and its induced CR-structures. The twistor spaces of the Lorentzian space forms $\R^4_1, \Di{S}^4_1$ and $\Di{H}^4_1$ are explicitly discussed. The given twistor construction is applied to surface theory in Lorentzian 4-spaces. Immersed spacelike surfaces in a Lorentzian 4-space with special geometric properties like semi-umbilic surfaces and surfaces that have mean curvature of vanishing lengths correspond to holomorphic curves in the Lorentzian twistor space. For the Lorentzian space forms $\R^4_1, \Di{S}^4_1$ and $\Di{H}^4_1$ those surfaces are explicitly constructed and classified.

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