Recommended by Christian Corda We obtain a height estimate concerning to a compact spacelike hypersurface Σ n immersed with constant mean curvature H in the anti-de Sitter space H n1 1 , when its boundary ∂Σ is contained into an umbilical spacelike hypersurface of this spacetime which is isometric to the hyperbolic space H n. Our estimate depends only on the value of H and on the geometry of ∂Σ...

It is proved that on a real hypersurface $M$ in the complex two-plane Grassmannian $G_2(\mathbb {C}^{m+2})$, structure Lie operator parallel if and only an open part of tube around totally geodesic {C}^{m+1})$ $G_2(\ma

We study umbilic (space-like) submanifolds of pseudo-Riemannian space forms, then define totally semi-umbilic space-like submanifold of pseudo Euclidean space and relate this notion to umbilicity. Finally we give characterization of total semi-umbilicity for space-like submanifolds contained in pseudo sphere or pseudo hyperbolic space or the light cone.A pseudo-Riemannian submanifold M in (a...

Let M be a compact (two-sided) minimal hypersurface in a Riemannian manifold M n+1 . It is a simple fact that if M has positive Ricci curvature then M cannot be stable (i. e. its Jacobi operator L has index at least one). If M = S(1) is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator. We prove that if M is the real projective space RP = Sn+1(1)/{±...

By establishing some differential geometry theory on the 1-lightlike surfaces, we show several geometric properties of the 1-lightlike surfaces which are completely different from non-lightlike surfaces. Based on these theories, we consider the singularities of the 1-lightlike surfaces in semiEuclidean 4-space with index two as an application of the theory of Legendrian singularities. We charac...

In this paper, by the Gauss equation of the induced Chern connection for Finsler submanifolds, we prove that if M is an umbilical hypersurface of a Minkowski space (V , F ), then either M is a Riemannian space form or a locally Minkowski space. AMS subject classifications: 53C60, 53C40

We study lightlike hypersurfaces of a semi-Riemannian product manifold. We introduce a class of lightlike hypersurfaces called screen semi-invariant lightlike hypersurfaces and radical antiinvariant lightlike hypersurfaces. We consider lightlike hypersurfaces with respect to a quartersymmetric nonmetric connection which is determined by the product structure. We give some equivalent conditions ...

In the present paper we study lightlike submanifolds of almost paracontact metric manifolds. We define invariant lightlike submanifolds. We study radical transversal lightlike submanifolds of para-Sasakian manifolds and investigate the geometry of distributions. Also we introduce a general notion of paracontact Cauchy-Riemann (CR) lightlike submanifolds and we derive some necessary and sufficie...

In this paper, the generalized helical hypersurfaces x=x(u,v,w) with a time-like axis in Minkowski spacetime E14 are considered. The first and second fundamental form matrices, Gauss map, shape operator matrix of x calculated. Moreover, curvatures hypersurface obtained by using Cayley–Hamilton theorem. umbilical conditions for given. Finally, Laplace–Beltrami is presented E14.

We study the problem of finding constant mean curvature graphs over a domain of a totally geodesic hyperplane and an equidistant hypersurface Q of hyperbolic space. We find the existence of graphs of constant mean curvature H over mean convex domains ⊂ Q and with boundary ∂ for −H∂ < H ≤ |h|, where H∂ > 0 is the mean curvature of the boundary ∂ . Here h is the mean curvature respectively of the...