Related papers: Geodesic completeness of some Lorentzian simple Li…

We study 4-dimensional simply connected Lie groups $G$ with left-invariant Riemannian metric $g$ admitting non-trivial conformal Killing 2-forms. We show that either the real line defined by such a form is invariant under the group action,…

In this paper, we investigated the behavior of left-invariant conformal vector fields on Lie groups with left-invariant pseudo-Riemannian metrics. First of all, we prove that conformal vector fields on pseudo-Riemannian unimodular Lie…

An important, if relatively less well known aspect of the singularity theorems in Lorentzian Geometry is to understand how their conclusions fare upon weakening or suppression of one or more of their hypotheses. Then, theorems with modified…

In this paper, we study the set of homogeneous geodesics of a leftinvariant Finsler metric on Lie groups. We first give a simple criterion that characterizes geodesic vectors. As an application, we study some geometric properties of…

Trajectories of light rays in a static spacetime are described by unparametrised geodesics of the Riemannian optical metric associated with the Lorentzian spacetime metric. We investigate the uniqueness of this structure and demonstrate…

To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi Riemannian metrics on…

We give a complete local classification of all Riemannian 3-manifolds $(M,g)$ admitting a nonvanishing Killing vector field $T$. We then extend this classification to timelike Killing vector fields on Lorentzian 3-manifolds, which are…

We study the mechanisms of the non properness of the action of the group of diffeomorphisms on the space of Lorentzian metrics of a compact manifold. In particular, we prove that nonproperness entails the presence of lightlike geodesic…

In this paper, we investigate left-invariant geodesic orbit metrics on connected simple Lie groups, where the metrics are formed by the structures of generalized flag manifolds. We prove that all these left-invariant geodesic orbit metrics…

We address the problem of finding conditions under which a compact Lorentzian manifold is geodesically complete, a property, which always holds for compact Riemannian manifolds. It is known that a compact Lorentzian manifold is geodesically…

We consider the three-dimensional Heisenberg group, equipped with any left-invariant metric, either Lorentzian or Riemannian. We completely classify their affine vector fields and investigate their relationship with Killing vector fields…

We study the existence and cardinality of normal geodesics of different causal types on H(eisenberg)-type quaternion group equipped with the sub-Lorentzian metric. We present explicit formulas for geodesics and describe reachable sets by…

Let $G/K$ be an orbit of the adjoint representation of a compact connected Lie group $G$, $\sigma$ be an involutive automorphism of $G$ and $\tilde G$ be the Lie group of fixed points of $\sigma$. We find a sufficient condition for the…

Let G be a Lie group equipped with a left-invariant semi-Riemannian metric. Let K be a semisimple subgroup of G generating a left-invariant conformal foliation F of codimension two on G. We then show that the foliation F is minimal. This…

We study the geometry of Lie groups $G$ with a continuous Finsler metric, assuming the existence of a subgroup $K$ such that the metric is right-invariant for the action of $K$. We present a systematic study of the metric and geodesic…

If $G$ is a compact Lie group endowed with a left invariant metric $g$, then $G$ acts via pullback by isometries on each eigenspace of the associated Laplace operator $\Delta_g$. We establish algebraic criteria for the existence of left…

We consider a compact manifold $(M,\mathfrak{F})$ with a foliation $\mathfrak{F}$, and a smooth affine connection $\nabla$ on the tangent bundle of the foliation $T\mathfrak{F}$. We introduce and study a foliated completeness problem.…

Let N be a nilpotent Lie group and let S be an invariant geometric structure on N (cf. symplectic, complex or hypercomplex). We define a left invariant Riemannian metric on N compatible with S to be "minimal", if it minimizes the norm of…

Let $G$ be a Lie group equipped with a left-invariant Riemannian metric. Let $K$ be a semisimple and normal subgroup of $G$ generating a left-invariant conformal foliation $\F$ of on $G$. We then show that the foliation $\F$ is Riemannian…

Given a Lorentzian manifold $(M,g_L)$ and a timelike unitary vector field $E$, we can construct the Riemannian metric $g_R=g_L+2\omega\otimes\omega$, being $\omega$ the metrically equivalent one form to $E$. We relate the curvature of both…