Holonomy of Weyl connections in Lorentzian space classified.
problem Classifying holonomy algebras of Weyl connections in Lorentzian signature.
method Classification through construction of examples of Weyl connections with all possible holonomy algebras.
result Examples of Weyl connections with all possible holonomy algebras constructed.
New connections found with specific torsion properties.
problem Understanding metric connections with specific torsion properties.
method Described Lorentzian manifolds with metric connections having parallel, skew-symmetric torsion.
result Found new Lorentzian manifolds with metric connections having parallel, skew-symmetric torsion.
The paper classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying algebraic Ricci solitons on three-dimensional Lorentzian Lie groups.
method Computed canonical and Kobayashi-Nomizu connections and their curvatures; defined algebraic Ricci solitons.
result Classified algebraic Ricci solitons on specific Lorentzian Lie groups.
Classifies Ricci collineations on specific 3D Lorentzian Lie groups.
problem Classifying Ricci collineations on specific Lie groups.
method Classifying based on canonical and Kobayashi-Nomizu connections.
result Results in classification of Ricci collineations.
Classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying Ricci solitons on three-dimensional Lorentzian Lie groups.
method Examined canonical and perturbed canonical connections, as well as Kobayashi-Nomizu connections and perturbed Kobayashi-Nomizu connections.
result Classified affine Ricci solitons on three-dimensional Lorentzian Lie groups with product structure.
The paper classifies Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures.
problem Classifying Lorentzian Lie groups based on specific tensor properties.
method Classification of three-dimensional Lorentzian Lie groups based on Ricci tensors and quasi-statistical structures associated with different affine connections.
result The paper classifies three-dimensional Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures associated with Bott, canonical, and Kobayashi-Nomizu connections.
The paper classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying Ricci solitons on Lorentzian Lie groups.
method Computed Bott connections and their curvature; classified Ricci solitons.
result Classification of Ricci solitons on three-dimensional Lorentzian Lie groups.
The paper explores Lorentzian connections with parallel skew torsion.
problem Understanding metric connections with parallel skew-symmetric torsion in Lorentzian signature.
method Analyzing holonomy algebras, torsion, and curvature; constructing examples; classifying homogeneous spaces.
result Complete classification of Lorentzian naturally reductive homogeneous spaces in low dimensions.
Study how algebraic conditions on isotropy group affect Lorentzian homogeneous space geometry.
problem Understand how algebraic conditions on isotropy group affect the geometry and curvature of Lorentzian homogeneous spaces.
method Prove that a Lorentzian locally homogeneous space is locally isometric to a plane wave if it admits an Ambrose--Singer connection with indecomposable, non-irreducible holonomy.
result Generalize existing results about Lorentzian homogeneous spaces with irreducible isotropy and prove results about Lorentzian connections with parallel torsion and 2-symmetric connections.
Defines and classifies algebraic Schouten solitons in 3D Lorentzian Lie groups.
problem Classifying solitons in 3D Lorentzian Lie groups.
method Defined algebraic Schouten solitons and classified them for specific connections.
result Classified algebraic Schouten solitons for various connections on 3D Lorentzian Lie groups.
Study finds all conformal Ricci collineations on specific 3D Lorentzian groups.
problem Identifying conformal Ricci collineations on three-dimensional Lorentzian Lie groups.
method Analysis of Levi-Civita connection on specific Lie groups.
result Determined all conformal Ricci collineations associated with the Levi-Civita connection.
Study finds all Ricci collineations for specific connections on 3D Lorentzian groups.
problem Identifying Ricci collineations for specific connections on 3D Lorentzian Lie groups.
method Examined left-invariant Ricci collineations associated with Bott connections on three-dimensional Lorentzian Lie groups.
result Determined all left-invariant Ricci collineations associated with the Bott connection.
The paper classifies solitons on specific Lie groups.
problem Classifying solitons on three-dimensional Lorentzian Lie groups.
method Computing Wanas tensor and defining algebraic Wanas solitons.
result Classification of algebraic Wanas solitons on specific Lie groups.
Classifies 1-connected Lorentzian manifolds with essential conformal groups.
problem Classifying Lorentzian manifolds with specific conformal groups.
method Proves existence of a metric making the manifold homogeneous and plane wave.
result Completes the classification of 1-connected Lorentzian manifolds with transitive conformal groups.
The study classifies special flows on specific geometric groups.
problem Classifying special flows on specific geometric groups.
method Classification of Left-invariant Ricci collineations associated to Yano connections on three-dimensional Lorentzian Lie groups.
result Results in classifying these flows.
In this paper, a survey of the recent results about the classification of the connected holonomy groups of the Lorentzian manifolds is given. A simplification of the construction of the Lorentzian metrics with all possible connected holonomy groups is obtained. As the applications, the Einstein equation, Lorentzian man…
New types of Ricci solitons found in 4D Lorentzian geometry.
problem Understanding Ricci solitons in Lorentzian geometry.
method Analyzing four-dimensional Lie groups for left-invariant Lorentz metrics.
result Any connected and simply connected 4D Lie group admits a left-invariant Lorentz metric that is a Ricci soliton.
We prove that the conformal group of a closed, simply connected, real analytic Lorentzian manifold is compact. D'Ambra proved in 1988 that the isometry group of such a manifold is compact. Our result implies the Lorentzian Lichnerowicz Conjecture for real analytic Lorentzian manifolds with finite fundamental group. Thi…
We study sub-Riemannian and sub-Lorentzian geometry on the Lie group $\SU(1,1)$ and on its universal cover $\CSU(1,1)$. In the sub-Riemannian case we find the distance function and completely describe sub-Riemannian geodesics on both $\SU(1,1)$ and $\CSU(1,1)$, connecting two fixed points. In particular, we prove that …
Study classifies Einstein-Yang-Mills spaces in 4D symmetric spaces.
problem Classifying Lorentzian symmetric spaces with Einstein-Yang-Mills properties.
method Classification based on invariant metric connections and diagonal metrics.
result Four-dimensional symmetric spaces with nontrivial isotropy groups are classified.
Study on averaging geometric structures in Finsler spaces with Lorentzian signature.
problem Averaging geometric structures in Finsler spaces with Lorentzian signature.
method Definition of an average connection without using the timelike vector field.
result No direct relation between the two averaged objects.
Derives Levi-Civita connection formulas for specific geometries.
problem Determining geometric invariants of Lorentzian manifolds.
method Explicitly derives Christoffel symbols in terms of adapted frame fields.
result Formulas for geometric invariants of Lorentzian manifolds.
Study left invariant Lorentzian metrics on 3D non-unimodular Lie groups.
problem Classify and analyze left invariant Lorentzian metrics on 3D non-unimodular Lie groups.
method Classify metrics up to automorphism, study curvature functions.
result Obtain Ricci operator, scalar curvature, and sectional curvatures as functions of metrics.
Characterizes Lorentzian manifolds with semi-symmetric metric connections.
problem Characterizing Lorentzian manifolds with specific metric connections.
method Analyzing semi-symmetric metric connections with vanishing curvature and recurrent torsion.
result Establishes conditions for perfect fluid and generalized Robertson-Walker spacetimes.
This paper explores Lorentzian manifolds with specific connections and their symmetries.
problem Characterizing Lorentzian manifolds with concircularly semi-symmetric metric connections.
method Investigates the properties of Lorentzian manifolds equipped with a concircularly semi-symmetric metric connection under specific conditions.
result Derives necessary and sufficient conditions for the manifold to be Einstein and proves that a perfect fluid space-time with a semi-symmetric metric P-connection is Ricci pseudo-symmetric manifold of constant type. Modeling wormhole creation without singularities in relativity.
problem Creating wormholes without singularities in classical relativity.
method Topological surgery and Morse theory to construct a nonsingular wormhole.
result Wormholes can be created nonsingularly in classical relativity.
Classifies a specific type of Lie groups related to Einstein geometry.
problem Classifying Einstein Lorentzian 3-nilpotent Lie groups with 1-dimensional nondegenerate center.
method Complete classification through mathematical analysis.
result A full classification of the specified Lie groups.
We describe the structure of d-dimensional homogeneous Lorentzian G-manifolds M=G/H of a semisimple Lie group G. Due to a result by N. Kowalsky, it is sufficient to consider the case when the group G acts properly, that is the stabilizer H is compact. Then any homogeneous space G/Hˉ with a smaller gro…
Identifies Lorentzian locally symmetric spaces where Calabi operator suffices to determine Killing operator range.
problem Determining when the Calabi operator can identify the range of the Killing operator in Lorentzian locally symmetric spaces.
method Developed criteria for a connection to be in the range of a connection, applied to the Killing connection.
result For indecomposable spaces, the Calabi operator suffices to identify the range of the Killing operator; for products, it fails.
The paper describes and analyzes maximal Lorentzian surfaces with a Killing field, proving completeness criteria.
problem Characterizing and understanding completeness of Lorentzian surfaces with a Killing field.
method Global description and analysis of surfaces, completeness criteria involving topology and geometry.
result Completeness of surfaces is equivalent to null completeness under bounded curvature hypothesis.
We call a connected Lie group endowed with a left-invariant Lorentzian flat metric Lorentzian flat Lie group. In this Note, we determine all Lorentzian flat Lie groups admitting a timelike left-invariant Killing vector field. We show that these Lie groups are 2-solvable and unimodular and hence geodesically complete. M…
We investigate connections between pairs of (pseudo-)Riemannian metrics whose sum is a (tensor) product of a covector field with itself. A bijective mapping between the classes of Euclidean and Lorentzian metrics is constructed as a special result. The existence of such maps on a differentiable manifold is discussed. S…
We study the geometry of compact Lorentzian manifolds that admit a somewhere timelike Killing vector field, and whose isometry group has infinitely many connected components. Up to a finite cover, such manifolds are products (or amalgamated products) of a flat Lorentzian torus and a compact Riemannian (resp., lightlike…
A method constructs tractor conformal bundles for spacelike submanifolds in Lorentzian manifolds.
problem Characterize conditions for a tractor conformal bundle to be standard and normal.
method Introduce a canonical construction of a tractor conformal bundle and characterize conditions for it to be standard and normal.
result Characterizes conditions for a tractor conformal bundle to be standard and normal.
We provide a classification of ts-invariant sub-Lorentzian structures on 3 dimensional contact Lie groups. Our approach is based on invariants arising form the construction of a normal Cartan connection.
Some results related to the causality of compact Lorentzian manifolds are proven: (1) any compact Lorentzian manifold which admits a timelike conformal vector field is totally vicious, and (2) a compact Lorentzian manifold covered regularly by a globally hyperbolic spacetime admits a timelike closed geodesic, if some n…
The invariant metric affine connections on Berger spheres which are Einstein with skew torsion are determined in both Riemannian and Lorentzian signature. Expressions of such connections are explicitly given. In particular, every Berger sphere with Lorentzian signature admits invariant metric affine connections Einstei…
It is explained how to find the de~Rham decomposition of a Riemannian manifold and the Wu decomposition of a Lorentzian manifold. For that it is enough to find parallel symmetric bilinear forms on the manifold, and do some linear algebra. This result will allow to compute the connected holonomy group of an arbitrary Ri…
The fermionic signature operator is analyzed on globally hyperbolic Lorentzian surfaces. The connection between the spectrum of the fermionic signature operator and geometric properties of the surface is studied. The findings are illustrated by simple examples and counterexamples.
The present study initially identify the generalized symmetric connections of type (α,β), which can be regarded as more generalized forms of quarter and semi-symmetric connections. The quarter and semi-symmetric connections are obtained respectively when (α,β)=(1,0) and (α,β)=(0,1). Taking that into account, a ne…
Study homogeneous Lorentzian manifolds under reductive Lie groups, reducing descriptions to semisimple groups.
problem Characterize homogeneous Lorentzian manifolds under reductive Lie groups.
method Analyze manifolds M=G/L for connected reductive Lie groups G and reductive subgroup L; focus on totally reducible isotropy representations. result Homogeneous Lorentzian manifolds reduce to semisimple Lie groups, and are reductive.
Spaces of polynomials are shown to be Euclidean balls.
problem Understanding the geometry of Lorentzian and real stable polynomials.
method Refined connection between symmetric exclusion process and polynomial geometry.
result Spaces of Lorentzian and real stable polynomials are homeomorphic to closed Euclidean balls.
In this note, we construct new examples of Lorentzian Sasaki-Einstein (LSE) metrics on Smale manifolds M. It has already been established in \cite{Gmz2} that such metrics exist on the so-called torsion free Smale manifolds, i.e. the k-fold connected sum of S2×S3. Now, we show that LSE metrics exist on…
We propose a formulation of a Lorentzian quantum geometry based on the framework of causal fermion systems. After giving the general definition of causal fermion systems, we deduce space-time as a topological space with an underlying causal structure. Restricting attention to systems of spin dimension two, we derive th…
In abstract Yang-Mills theory the standard instanton construction relies on the Hodge star having real eigenvalues which makes it inapplicable in the Lorentzian case. We show that for the affine connection an instanton-type construction can be carried out in the Lorentzian setting. The Lorentzian analogue of an instant…
The study examines a semi-symmetric metric connection in perfect fluid space-time and phantom barriers.
problem Investigating the properties of semi-symmetric metric connections in perfect fluid space-time.
method Using concircularly semi-symmetric metric connections, the study derives conditions for quasi-Einstein manifolds and examines the scalar curvature of perfect fluid space-times.
result The study proves that in a perfect fluid space-time, the scalar curvature is constant and represents a phantom barrier.
The Oscillator Groups,$\G_λ,$ 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 aff…
Clarifies definitions of global hyperbolicity in various spaces.
problem Clarifying terminology in recent literature on global hyperbolicity.
method Comparing definitions in Lorentzian length spaces, optimal transport, and topological preordered spaces.
result The causal relation is a closed order and preserves compactness in all cases.