We study connected orientable spacelike hypersurfaces $x:M^{n}rightarrowM_q^{n+1}(c)$, isometrically immersed into the Riemannian or Lorentzian space form of curvature $c=-1,0,1$, and index $q=0,1$, satisfying the condition $~L_kx=Ax+b$,~ where $L_k$ is the $textit{linearized operator}$ of the $(k+1)$-th mean curvature $H_{k+1}$ of the hypersurface for a fixed integer $0leq k

A classical result in Lorentzian geometry states that a strongly causal spacetime is globally hyperbolic if and only if the Lorentzian distance is finite valued for every metric choice in the conformal class. It is proven here that a non-total imprisoning spacetime is causally simple if and only if for every metric choice in the conformal class the Lorentzian distance is continuous wherever it ...

We discuss a general theory of Lorentzian Kac–Moody algebras which should be a hyperbolic analogy of the classical theories of finite-dimensional semisimple and affine Kac–Moody algebras. First examples of Lorentzian Kac–Moody algebras were found by Borcherds. We consider general finiteness results about the set of Lorentzian Kac–Moody algebras and the problem of their classification. As an exa...

We introduce and solve a family of discrete models of 2D Lorentzian gravity with higher curvature weight, which possess mutually commuting transfer matrices, and whose spectral parameter interpolates between flat and curved space-times. We further establish a one-toone correspondence between Lorentzian triangulations and directed Random Walks. This gives a simple explanation why the Lorentzian ...

We give a short and simple proof that the Lorentzian 10j symbol, which forms a key part of the Barrett-Crane model of Lorentzian quantum gravity, is finite. The argument is very general, and applies to other integrals. For example, we show that the Lorentzian and Riemannian causal 10j symbols are finite, despite their singularities. Moreover, we show that integrals that arise in Cherrington’s w...

The geometry of manifolds endowed with geometrical structures has been intensively studied, and several important results have been published. In this paper, we deal with manifolds having a Lorentzian concircular structure LCS n-manifold 1–3 see Section 2 for detail . The study of the Lorentzian almost paracontact manifold was initiated by Matsumoto in 4 . Later on, several authors studied the ...

Fruitful ideas on how to quantize gravity are few and far between. In this paper, we give a complete description of a recently introduced non-perturbative gravitational path integral whose continuum limit has already been investigated extensively in d < 4, with promising results. It is based on a simplicial regularization of Lorentzian space-times and, most importantly, possesses a well-defined...

Lorentzian manifolds admitting a Killing vector field have been studied in the literature from different points of view. Functional analysis, proper actions of Lie groups or Bochner’s technique in Differential Geometry, have been some of the very different tools involved to study them. The purpose of this paper is to review some of their properties, pointing out several techniques and supplying...

It is shown that the temperature may be quantized in some spacetime due to the periodically topological structure of the Euclidean section. The quanta of the temperature is the Hawking-Unruh temperature. In the Euclidean quantum theory physical functions in the Lorentzian section of a complexified spacetime are assumed to be analytically continued from some functions in the Euclidean section [1...