Related papers: Robinson Manifolds as the Lorentzian Analogs of He…
The classifications of holonomy groups in Lorentzian and in Euclidean signature are quite different. A group of interest in Lorentzian signature in n dimensions is the maximal proper subgroup of the Lorentz group, SIM(n-2). Ricci-flat…
Extended recurrent pseudo-Riemannian manifolds were introduced by Mileva Prvanovic'. We reconsider her work in the light of recent results and show that the manifold is conformally flat, and it is a space of quasi-constant curvature. We…
Second-order symmetric Lorentzian spaces, that is to say, Lorentzian manifolds with vanishing second derivative of the curvature tensor R, are characterized by several geometric properties, and explicitly presented. Locally, they are a…
We investigate harmonic maps in the context of isometric embeddings when the target space is Ricci-flat and has codimension one. With the help of the Campbell-Magaard theorem we show that any $n$-dimensional ($n\geqslant 3$) Lorentzian…
An almost Robinson structure on an $n$-dimensional Lorentzian manifold $(\mcM,g)$, where $n=2m+\epsilon$, $\epsilon \in \{ 0 ,1 \}$, is a complex $m$-plane distribution $\mcN$ that is totally null with respect to the complexified metric,…
The classification of Riemannian manifolds by the holonomy group of their Levi-Civita connection picks out many interesting classes of structures, several of which are solutions to the Einstein equations. The classification has two parts.…
Riemannian Manifolds may be $C^{1,1}$ and the geometry of these manifolds is investigated in \cite{Groah1}. Here, a similar analysis is given for pseudohermitian, torsion-free manifolds whereby, instead of assuming that the metric is…
In this paper we investigate the problem of non-analytic embeddings of Lorentzian manifolds in Ricci-flat semi-Riemannian spaces. In order to do this, we first review some relevant results in the area, and then motivate both the…
The non-existence of non-trivial conformally symmetric manifolds in the three-dimensional Riemannian setting is shown. In Lorentzian signature, a complete local classification is obtained. Furthermore, the isometry classes are examined.
We classify the Ricci flat Lorentzian $n$-manifolds satisfying three particular conditions, encoding and combining some crucial features of the Kerr metrics and the Robinson-Trautman optical structures. We prove that: (a) If $n>4$, there is…
The holonomy algebras of Einstein not Ricci-flat pseudo-Riemannian manifolds of arbitrary signature are classified. As illustrating examples, the cases of Lorentzian manifolds, pseudo-Riemannian manifolds of signature $(2,n)$ and the…
The geodesic orbit property has been studied intensively for Riemannian manifolds. Geodesic orbit spaces are homogeneous and allow simplifications of many structural questions using the Lie algebra of the isometry group. Weakly symmetric…
We consider general relativity with cosmological constant minimally coupled to the electromagnetic field and assume that the four-dimensional space-time manifold is a warped product of two surfaces with Lorentzian and Euclidean signature…
Motivated by the application to spacetimes of general relativity we investigate the geometry and regularity of Lorentzian manifolds under certain curvature and volume bounds. We establish several injectivity radius estimates at a point or…
Sub-Riemannian Geometry is proved to play an important role in many applications, e.g., Mathematical Physics and Control Theory. The simplest example of sub-Riemannian structure is provided by the 3-D Heisenberg group. Sub-Riemannian…
Let $M$ be a connected, simply connected, oriented, closed, smooth four-manifold which is spin (or equivalently having even intersection form) and put $M^\times:=M\setminus\{{\rm point}\}$.In this paper we prove that if $X^\times$ is a…
A natural one codimension isometric embedding of each $(n+1)$-dimensional spherical Robertson-Walker (RW) spacetime $I\times_f \mathbb{S}^n$ in $(n+2)$-dimensional Lorentz-Minkowski spacetime $\mathbb{L}^{n+2}$ permits to contemplate…
The n-dimensional Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor (2-symmetric spacetimes) are characterized and classified. The main result is that either they are locally symmetric or they have a…
We present several new space-periodic solutions of the static vacuum Einstein equations in higher dimensions, both with and without black holes, having Kasner asymptotics. These latter solutions are referred to as gravitational solitons.…
In this work we study riemannian metrics on flag manifolds adapted to the symmetries of these homogeneous nonsymmetric spaces. We first introduce the notion of riemannian $\Gamma $-symmetric space when $\Gamma $ is a general abelian finite…