In this paper we modify the maximum principal of (Galloway, 2000) for totally geodesic null hypersurfaces by proving a geometric maximum principle which obeys mean curvature inequalities of a family of totally umbilical null hypersurfaces of a spacetime manifold (Theorem 6). As a physical interpretation we show that, in particular, for a prescribed class of spacetimes the geometric inequality o...

In this paper we study n-dimensional compact minimal submanifolds in S with scalar curvature S satisfying the pinching condition S > n(n − 2). We show that for p ≤ 2 these submanifolds are totally geodesic (cf. Theorem 3.2 and Corollary 3.1). However, for codimension p ≥ 2, we prove the result under an additional restrictions on the curvature tensor corresponding to the normal connection (cf. T...

We introduce a combinatorial curvature flow for piecewise constant curvature metrics on compact triangulated 3-manifolds with boundary consisting of surfaces of negative Euler characteristic. The flow tends to find the complete hyperbolic metric with totally geodesic boundary on a manifold. Some of the basic properties of the combinatorial flow are established. The most important one is that th...

The aim of the present paper is to study holomorphically planar conformal vector fields(HPCV) on almost alpha-cosymplectic (k,m)-spaces. This done assuming various conditions such as i) U pointwise collinear with xi ( in this case integral manifold distribution D totally geodesic or umbilic), ii) M has a constant xi-sectional curvature (under condition (or umbilic) isometric sphere S2n+1(pc) ra...

Abstract A well-known application of the Raychaudhuri equation shows that, under geodesic completeness, totally null hypersurfaces are unique which satisfy that Ricci curvature is nonnegative in direction. The proof this fact based on a direct analysis differential inequality. In paper, we show, without assuming an inequality involving squared mean and compact three-dimensional hypersurface als...

Let M denote a compact real hyperbolic manifold with dimensionm 2: 5 and sectional curvature K = I , and let 1: be an exotic sphere ofdimension m. Given any small number t5 > 0 , we show that there is a finitecovering space M of M satisfying the following properties: the connectedsum M#1: is not diffeomorphic to M, but it is homeomorphic to M; M#1:supports a Riemannian metri...

We consider Gromov–Thurston examples of negatively curved nmanifolds which do not admit metrics of constant sectional curvature. We show that for each n ≥ 4 some of the Gromov–Thurston manifolds admit strictly convex real–projective structures.