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

Conformal and Grassmann structures

The main results on the theory of conformal and almost Grassmann structures are presented. The common properties of these structures and also the differences between them are outlined. In particular, the structure groups of these structures and their differential prolongations are found. A complete system of geometric objects of the almost Grassmann structure totally defining its geometric structure is determined. The vanishing of these objects determines a locally Grassmann manifold. It is proved that the integrable almost Grassmann structures are locally Grassmann. The criteria of semiintegrability of almost Grassmann structures is proved in invariant form.Comment: LaTeX, 25 page