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Sfb 288 Differential Geometry and Quantum Physics |
Abstract for Sfb Preprint No. 370
Twistorial constructions of spacelike surfaces in Lorentzian 4-manifolds Felipe Leitner
A construction of J. Eells and S. Salamon [ES85] uses Riemannian twistor theory for the investigation of minimal surfaces in Riemannian 4-manifolds. In this paper the twistor space and the Grassmannian fibre bundle of a Lorentzian 4-space with natural almost optical structures and its induced CR-structures are studied. The twistor spaces of the Lorentzian space forms $Di{R}^4_1$, $Di{S}^4_1$ and $Di{H}^4_1$ are explicitly discussed. Similar as in Riemannian geometry, the given twistor construction is applied to surface theory in Lorentzian 4-spaces. Immersed surfaces with special geometric properties like semi-umbilic and (semi)-stationary spacelike surfaces correspond to holomorphic curves in the twistor space. For the Lorentzian space forms $Di{R}^4_1$, $Di{S}^4_1$ and $Di{H}^4_1$ those surfaces are explicitly constructed and classified
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