160 research outputs found
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
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