‎let $m^n$ be an $n(ngeq 3)$-dimensional complete connected and‎ ‎oriented spacelike hypersurface in a de sitter space or an anti-de‎ ‎sitter space‎, ‎$s$ and $k$ be the squared norm of the second‎ ‎fundamental form and gauss-kronecker curvature of $m^n$‎. ‎if $s$ or‎ ‎$k$ is constant‎, ‎nonzero and $m^n$ has two distinct principal‎ ‎curvatures one of which is simple‎, ‎we obtain some‎ ‎charact...

We obtain necessary and sufficient conditions for an initial data set the vacuum conformal Einstein field equations to give rise a spacetime development in possession of Killing spinor. The fact that are used our derivation allows possibility hypersurface S intersecting non-trivially with (or even being subset of) null infinity I. For conciseness, these derived assuming is spacelike. Hence, par...

In this study, the parametric equation of hypersurfaces passing through spacelike and timelike curves that has non-null Frenet vectors in E_1^4 Lorentz Minkowski space was expressed with help curve is frame. Furthermore, hypersurface families were created by giving necessary sufficient conditions so on common isoasymptotic hypersurface. Then, these obtained deviation marching-scale functions si...

‎Let $M^n$ be an $n(ngeq 3)$-dimensional complete connected and‎ ‎oriented spacelike hypersurface in a de Sitter space or an anti-de‎ ‎Sitter space‎, ‎$S$ and $K$ be the squared norm of the second‎ ‎fundamental form and Gauss-Kronecker curvature of $M^n$‎. ‎If $S$ or‎ ‎$K$ is constant‎, ‎nonzero and $M^n$ has two distinct principal‎ ‎curvatures one of which is simple‎, ‎we obtain some‎ ‎charact...

The causal boundary of string propagation – defined as the hypersurface in loop space bordering the timelike(spacelike) domains in which two successive measurements of the string field do(do not) interfere with one another – is argued to be 0 = dσ δX(σ) 2 = ∞ ℓ=−∞ δx µ −ℓ δx µ ℓ. Some possible consequences are discussed.

What is the shape of space in a spacetime? One way of addressing this issue is to consider edgeless spacelike submanifolds of the spacetime. An alternative is to foliate the spacetime by timelike curves and consider the quotient obtained by identifying points on the same timelike curve. In this article we investigate each of these notions and obtain conditions such that it yields a meaningful s...

The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit “hidden symmetry” from conformal KillingYano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimensiontwo submanifold with constant normalized null expansion (null mean curva...

The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit “hidden symmetry” from conformal KillingYano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimensiontwo submanifold with constant normalized null expansion (null mean curva...

Let Mn+p p (c) be n + p-dimensional connected semi-Riemannian manifold of constant curvature c whose index is p. It is called indefinite space form of index p. Let M be an n-dimensional Riemannian manifold immersed in Mn+p p (c). The semi-Riemannian metric of Mn+p p (c) induces the Riemannian metric of M , M is called a spacelike submanifold. Spacelike submanifolds in indefinite space form Mn+p...

Foliations by constant mean curvature hypersurfaces provide a possibility of defining a preferred time coordinate in general relativity. In the following various conjectures are made about the existence of foliations of this kind in spacetimes satisfying the strong energy condition and possessing compact Cauchy hypersurfaces. Recent progress on proving these conjectures under supplementary assu...