The main purpose of this paper is to give fundamental properties of real lightlike hypersurfaces of paraquaternionic manifolds and to prove the non-existence of real lightlike hypersurfaces in paraquaternionic space form under some conditions. AMS MATHEMATICS SUBJECT CLASSIFICATION. Primary 53C15, Secondary 53C42, 53C50.

We consider the totally nonfree (TNF) action of a groups and the corresponding adjoint invariant (AD) measures on the lattices of the subgroups of the given group. The main result is the description of all adjoint-invariant and TNF-measures on the lattice of subgroups of the infinite symmetric group SN. The problem is closely related to the theory of characters and factor representations of gro...

The aim of this paper is to show how we can induce contact structures, metric structures and Sasaki on a null hypersurface from rigging vector field. We give several explicit examples construction some obstructions its existence. For example, be introduced only zero mean curvature hypersurfaces, whereas in totally geodesic ones. In particular construct Kähler halo around an isolated horizon bla...

Let M be an n-dimensional submanifold in the simply connected space form F(c) with c + H > 0, where H is the mean curvature of M . We verify that if M(n ≥ 3) is an oriented compact submanifold with parallel mean curvature and its Ricci curvature satisfies RicM ≥ (n−2)(c+H2), then M is either a totally umbilic sphere, a Clifford hypersurface in an (n+1)-sphere with n = even, or CP ( 4 3 (c+H )) ...

The subject of limit curve theorems in Lorentzian geometry is reviewed. A general limit curve theorem is formulated which includes the case of converging curves with endpoints and the case in which the limit points assigned since the beginning are one, two or at most denumerable. Some applications are considered. It is proved that in chronological spacetimes, strong causality is either everywhe...

We establish the links between lightlike geometry and basics invariants of associated semi-Riemannian on r-lightlike submanifold constructed from a ambient. Then we some basic inequalities, involving scalar curvature shape operator coisotropic in manifold. Equality cases are also discussed.

In this paper we show that bending a finite volume hyperbolic d-manifold M along a totally geodesic hypersurface Σ results in a properly convex projective structure on M with finite volume. We also discuss various geometric properties of bent manifolds and algebraic properties of their fundamental groups. We then use this result to show in each dimension d Ê 3 there are examples finite volume, ...