On geometry of hypersurfaces of a pseudoconformal space of Lorentzian signature

There are three types of hypersurfaces in a pseudoconformal space C n 1 of Lorentzian signature: spacelike, timelike, and lightlike. These three types of hypersurfaces are considered in parallel. Spacelike hypersurfaces are endowed with a proper conformal structure, and timelike hypersurfaces are en...

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Bibliographic Details
Published in:Journal of geometry and physics 1998-06, Vol.26 (1), p.112-126
Main Authors: Akivis, M.A., Goldberg, V.V.
Format: Article
Language:English
Subjects:
Differential geometry
Hypersurfaces
Lorentzian signature
Pseudoconformal space
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Summary:There are three types of hypersurfaces in a pseudoconformal space C n 1 of Lorentzian signature: spacelike, timelike, and lightlike. These three types of hypersurfaces are considered in parallel. Spacelike hypersurfaces are endowed with a proper conformal structure, and timelike hypersurfaces are endowed with a conformal structure of Lorentzian type. Geometry of these two types of hypersurfaces can be studied in a manner that is similar to that for hypersurfaces of a proper conformal space. Lightlike hypersurfaces are endowed with a degenerate conformal structure. This is the reason that their investigation has special features. It is proved that under the Darboux mapping such hypersurfaces are transferred into tangentially degenerate ( n - 1)-dimensional submanifolds of rank n - 2 located on the Darboux hyperquadric. The isotropic congruences of the space C n 1 that are closely connected with lightlike hypersurfaces and their Darboux mapping are also considered.
ISSN:0393-0440
1879-1662
DOI:10.1016/S0393-0440(97)00041-7