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 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.Comment: LaTeX, 21 page

Differential systems associated with tableaux over Lie algebras

We give an account of the construction of exterior differential systems based on the notion of tableaux over Lie algebras as developed in [Comm. Anal. Geom 14 (2006), 475-496; math.DG/0412169]. The definition of a tableau over a Lie algebra is revisited and extended in the light of the formalism of the Spencer cohomology; the question of involutiveness for the associated systems and their prolongations is addressed; examples are discussed.Comment: 16 pages; to appear in: "Symmetries and Overdetermined Systems of Partial Differential Equations" (M. Eastwood and W. Miller, Jr., eds.), IMA Volumes in Mathematics and Its Applications, Springer-Verlag, New Yor

On Four-Dimensional Conformal Structures

. The conformal structures CO(4; 0); CO(1; 3) and CO(2; 2) are studied on a real manifold M; dimM = 4. On M isotropic fiber bundles E ff and E fi are constructed. These bundles are real for the CO(2; 2)-structure, and they satisfy the condition E ff = E fi for the CO(1; 3)-structure, and the conditions E ff = E ff ; E fi = E fi for the CO(4)-structure. The tensor C of conformal curvature splits into two subtensors C ff and C fi which are the curvature tensors of the bundles E ff and E fi , respectively. These subtensors satisfy the same conditions as the bundles E ff and E fi . Conformally semiflat and flat structures and their geometrical characteristics are studied. The principal 2-directions are defined, and conditions for their integrability are obtained. These investigations for the CO(1; 3)-structure are connected with Petrov's classification of Einstein's spaces. Although the paper is of a servey nature, it also contains some new results. 1. Four-dimensional conformal structur..

Lightlike Hypersurfaces On Manifolds Endowed With A Conformal Structure Of Lorentzian Signature

. The authors study the geometry of lightlike hypersurfaces on manifolds (M; c) endowed with a pseudoconformal structure c = CO(n \Gamma 1; 1) of Lorentzian signature. Such hypersurfaces are of interest in general relativity since they can be models of different types of physical horizons. On a lightlike hypersurface, the authors consider the fibration of isotropic geodesics and investigate their singular points and singular submanifolds. They construct a conformally invariant normalization of a lightlike hypersurface intrinsically connected with its geometry and investigate affine connections induced by this normalization. The authors also consider special classes of lightlike hypersurfaces. In particular, they investigate lightlike hypersurfaces for which the elements of the constructed normalization are integrable. 0 Introduction The pseudo-Riemannian manifolds (M; g) of Lorentzian signature play a special role in geometry and physics: they generate models of spacetime of general ..