The Oscillator Groups,Gλ, are the only solvable, non commutative, simply connected Lie groups to admit a Lorentzian bi-invariant metric. For these groups, we give sufficient conditions for a left-invariant pseudo-Riemannian metric to be complete, we determine the group of isometries, we exhibit a left-invariant affine structure and prove that it is not Lorentzian. As an application, we provide ...

The geometric maximum principle for smooth (spacelike) hypersurfaces, which is a consequence of Alexandrov’s [1] strong maximum for second order quasilinear elliptic operators, is a basic tool in Riemannian and Lorentzian geometry. In [2], extending earlier work of Eschenburg [7], a version of the geometric maximum principle in the Lorentzian setting was obtained for rough (C) spacelike hypersu...

A regular vacuum solution in 5D gravity on the principal bundle with the U (1) structural group is proposed as a 4D wormhole. This solution has two null hypersurfaces where an interchange of the sign of some 5D metric components happens. For a 4D observer living on the base of this principal bundle this is a wormhole with two asymptotically flat Lorentzian (Euclidean) spacetimes connected by a ...

Abstract: Completing the results achieved in a previous paper, we prove the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients in smooth D-dimensional Lorentzian manifolds. To this end, it is shown that, in any Lorentzian manifold, a sort of “local Wick rotation” of the metric can be performed provided the metric is a locally analytic function of the coordinates and the coordinates ar...

I introduce a closeness function between causal Lorentzian geometries of finite volume and arbitrary underlying topology. The construction uses the fact that some information on the manifolds and metrics is encoded in the partial order that the causal structure of each metric induces among points randomly scattered in the corresponding manifold with uniform, finite density according to the volu...

A maximal surface S with isolated singularities in a complete flat Lorentzian 3-manifold N is said to be entire if it lifts to a (periodic) entire multigraph S̃ in L. In addition, S is called of finite type if it has finite topology, finitely many singular points and S̃ is a finitely sheeted multigraph. Complete (or proper) maximal immersions with isolated singularities in N are entire, and entir...

This talk will deal to a certain extent with the natural continuation of some work in progress that was introduced last year at the Spanish Relativity Meeting in Salamanca 1, 2 and that has recently been published 3. The reason for investigating geodesic completeness of Lorentzian manifolds is twofold. From the mathematical point of view, there is no analogue for the Hopf-Rinow theorem that cha...

Kuchař has recently given a detailed analysis of the classical and quantum geometrodynamics of the Kruskal extension of the Schwarzschild black hole. In this paper we adapt Kuchař’s analysis to the exterior region of a Schwarzschild black hole with a timelike boundary. The reduced Lorentzian Hamiltonian is shown to contain two independent terms, one from the timelike boundary and the other from...

The purpose of the paper is to introduce the notion of generalized recurrent Lorentzian transSasakian manifold and study some of the properties of generalized recurrent and Ricci recurrent Lorentzian Trans-Sasakian manifolds.

In this note a proof is given for global existence and uniqueness of minimal Lorentzian surface maps from a cylinder into globally hyperbolic Lorentzian manifolds for given initial values up to the first derivatives.