The study characterizes pseudo-Riemannian foliations and their graphs.
problem Characterizing pseudo-Riemannian foliations and their graphs.
method Analyzing geodesics and metrics on foliation graphs.
result A unique pseudo-Riemannian metric exists on foliation graphs.
Study properties of semi-symmetric Lorentzian spaces, foliated manifolds.
problem Properties of semi-symmetric pseudo-Riemannian manifolds.
method Investigate foliated manifolds with Lorentzian metrics and analyze Ricci operator eigenvalues.
result Ricci operator has only real eigenvalues for Lorentzian metrics.
We investigate under which assumptions an orientable pseudo-Riemannian geodesic foliations by circles is generated by an S1-action. We construct examples showing that, contrary to the Riemannian case, it is not always true. However, we prove that such an action always exists when the foliation does not contain light…
The paper studies variations of metrics on foliated manifolds and finds solutions to specific actions.
problem Variations of metrics on foliated pseudo-Riemannian manifolds.
method Developed variation formulas and applied to Einstein-Hilbert type actions.
result Found solutions like twisted products, conformal submersions, and isoparametric foliations.
Study Lie foliation of Walker manifolds in pseudo-Riemannian geometry.
problem Characterize Lie foliations of Walker manifolds.
method Analyzes Walker manifolds with null parallel distributions to integrate to Lie foliations and studies their properties.
result Walker manifolds with null parallel distributions always integrate to Lie foliations, and the transverse holonomy coincides with the image of the holonomy morphism.
A necessary and sufficient condition for the leaves of a {\em non-degenerate} foliation of a pseudo-Riemannian manifold to be conformally flat is developed. The condition mimics the classical condition of the vanishing of the Weyl or Cotton tensor establishing the conformal flatness of a pseudo-Riemannian manifold in t…
Study properties of null string congruences in complex and real pseudo-Riemannian spaces.
problem Analyze properties of null string congruences in 4D spaces.
method Investigate the geometry of null strings and their congruences in complex and real pseudo-Riemannian spaces.
result Relations between null string congruences, Petrov-Penrose types, and Ricci tensor algebraic types are established.
In this paper a thorough study of the normal form and the first integrability conditions arising from {\em bi-conformal vector fields} is presented. These new symmetry transformations were introduced in {\em Class. Quantum Grav.}\textbf{21}, 2153-2177 and some of their basic properties were addressed there. Bi-conforma…
We prove Wadsley's theorem for foliations by closed non-lightlike geodesics. As an application we show that every pseudo-Riemannian and non-Riemannian 2-mainfold, all of whose time- or spacelike geodesics are closed, is diffeomorphic to S1×R. Further we show that every pseudo-Riemannian 2-manifold with index …
Defines indefinite Kasparov modules for non-elliptic operators.
problem Modeling non-symmetric and non-elliptic operators.
method Introduces indefinite Kasparov modules and proves their properties.
result Reversibility of the association between indefinite Kasparov modules and genuine Kasparov modules.
In this paper we continue the study of bi-conformal vector fields started in {\em Class. Quantum Grav.} {\bf 21} 2153-2177. These are vector fields defined on a pseudo-Riemannian manifold by the differential conditions $\lie P_{ab}=φP_{ab}$, $\lieΠ_{ab}=χΠ_{ab}$ where Pab, Πab are orthogonal and complementary…
The paper explores formulas and applications for mixed scalar curvature in multi-product manifolds.
problem Integral and variation formulas for mixed scalar curvature in multi-product manifolds.
method Generalizes results from pseudo-Riemannian almost product manifolds to multi-product structures.
result Generalizes formulas for mixed scalar curvature in multi-product manifolds.
The study finds obstructions for certain Weyl curvature tensors on manifolds.
problem Can manifolds admit metrics with purely electric or magnetic Weyl tensors?
method Analyzes algebraic curvature tensors and their Pontryagin classes on scalar product spaces.
result Obstructions to the existence of metrics with PE or PM Weyl tensors in top-degree cohomology.
We study conditions for the integrability of the distribution defined on a regular Poisson manifold as the orthogonal complement (with respect to some (pseudo)-Riemannian metric) to the tangent spaces of the leaves of a symplectic foliation. Examples of integrability and non-integrability of this distribution are provi…
We consider foliations of the whole three dimensional hyperbolic space H^3 by oriented geodesics. Let L be the space of all the oriented geodesics of H^3, which is a four dimensional manifold carrying two canonical pseudo-Riemannian metrics of signature (2,2). We characterize, in terms of these geometries of L, the sub…
Study on null-projectability of Levi-Civita connections in neutral metrics.
problem Characterizing projectability of Levi-Civita connections along null parallel distributions.
method Analyzing projectability of torsion-free connections along foliations on manifolds, focusing on neutral metric signatures and mid-dimensional distributions.
result Extension of Patterson and Walker's Riemann extension metrics to null parallel distributions of any dimension.
Study variational formulas for distribution geometry, finding critical metrics.
problem Analyzing the total mixed scalar curvature of a distribution.
method Developed variational formulas for extrinsic geometry, solved Euler-Lagrange equations.
result Found critical metrics related to various geometric properties.
Conditions for metallic structures to induce integrable distributions and geodesic invariance.
problem Conditions for metallic structures to induce integrable and geodesically invariant distributions.
method Analyzing tensor fields and adapted connections associated with metallic structures.
result Chen-type inequality for metallic distributions and conditions for metallic maps to preserve these distributions.
The paper studies biharmonic submanifolds in pseudo-Riemannian manifolds.
problem Characterizing biharmonic submanifolds in pseudo-Riemannian geometry.
method Derived biharmonic equations and proved properties of biharmonic submanifolds.
result Found constant mean curvature for pseudo-umbilical biharmonic submanifolds.
Study on pseudo-Riemannian submanifolds with specific geodesic properties.
problem Characterizing geodesic properties in pseudo-Riemannian submanifolds.
method Analyzing W-curves and normal sections on pseudo-Riemannian submanifolds.
result Necessary and sufficient conditions for normal sections to be W-curves.
The paper characterizes metallic pseudo-Riemannian manifolds using conjugate connections and tensor structures.
problem Characterizing metallic pseudo-Riemannian manifolds.
method Using conjugate connections and tensor structures, the paper derives new characterizations and conditions for these manifolds.
result A necessary and sufficient condition for a non-integrable metallic pseudo-Riemannian manifold to be a quasi metallic pseudo-Riemannian manifold is derived.
The paper presents new formulations of gauge and gravity theories using dynamical principal bundles.
problem Formulating gauge and gravity theories with a flexible principal bundle structure.
method Original variational formulations of Yang-Mills, Einstein's gravitation, and Kaluza-Klein theories with a dynamical principal bundle.
result The principal bundle structure and connection emerge from the dynamics, leading to solutions of Yang-Mills, Einstein-Cartan, or Yang-Mills-Einstein equations.
Study on pseudo-Riemannian algebraic Ricci solitons in 4D Lie groups.
problem Investigating conditions for pseudo-Riemannian algebraic Ricci solitons on 4D Lie algebras.
method Analyzing the algebraic Ricci soliton equation for each 4D Lie algebra.
result Complete description of pseudo-Riemannian algebraic Ricci solitons in dimension four.
We classify pseudo-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a pseudo-Riemannian manifold. Also, we obtain the classification of the pseudo-Riemannian submersions with (para-)complex connected totally geodesic fibres from a (para-)complex pseudo-hyperbolic sp…
Study on isometries in contact sub-pseudo-Riemannian geometry.
problem Understanding isometries in a specific geometric structure.
method Analyzing upper bounds and maximal dimensions of isometry groups.
result Maximal dimension of isometry groups is achieved for Heisenberg group structures.
Study on pseudo-Riemannian Bertrand manifolds finds no closed movement systems.
problem Exploring properties of pseudo-Riemannian Bertrand manifolds.
method Proof of properties and discussion of the theory.
result No pseudo-Riemannian completely Bertrand systems exist.
Establishing Hom-versions of Bochner theorems in pseudo-Riemannian Hom-Lie algebras
problem Killing vectors and Bochner-type theorems in pseudo-Riemannian Hom-Lie algebras
method Hom-versions of Bochner theorems
result Space of Killing vectors forms a totally geodesic Hom-Lie subalgebra
The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold (M,g) is closely related to the existence of a parallel 1-dimensional complex subbundle of the spinor bundle of (M,g). We characterize the following simply connected pseudo-Riemannian manifolds admitting such subbundles in terms of their…
We generalize for pseudo-Riemannian metrics a classical result of Gallot and Tanno and use it to reprove a recent result of Alekseevsky, Cortes, Galaev and Leistner that decomposable cones over complete closed pseudo-Riemannian manifolds do not exist.
New vector fields define group actions on pseudo-Riemannian spaces.
problem Investigate I-degenerate pseudo-Riemannian spaces. method Define Lie algebras of vector fields using Lie derivative of the metric and study their group actions.
result All known I-degenerate pseudo-Riemannian spaces have the defined vector fields. Classification of specific pseudo-Riemannian manifolds.
problem Classifying conformally flat generalized Ricci recurrent pseudo-Riemannian manifolds.
method Complete classification through mathematical analysis.
result Conformally flat generalized Ricci recurrent pseudo-Riemannian manifolds are either de Sitter or anti-de Sitter spacetimes.
Semisimple Lie groups act transitively on pseudo-Riemannian manifolds, making them flat.
problem Understanding actions of semisimple Lie groups on pseudo-Riemannian manifolds.
method Analyzing the pseudo-Riemannian Lichnerowicz conjecture in homogeneous settings.
result Compact pseudo-Riemannian manifolds on which a semisimple group acts conformally, essentially and transitively, are conformally flat.
Classifies flat pseudo-Riemannian spaces with specific structures.
problem Classifying homogeneous pseudo-Riemannian spaces with invariant structures.
method Classification based on invariant almost hyper-Hermitian structures and H-irreducible isotropy groups.
result All classified spaces are flat except in dimension 12.
Study shows solvable Lie groups act freely on compact pseudo-Riemannian manifolds.
problem Understanding the action of solvable Lie groups on compact pseudo-Riemannian manifolds.
method Analyzes the action of a solvable Lie group on a compact pseudo-Riemannian manifold and shows the group acts almost freely.
result The identity component of the isometry group of the manifold coincides with the group.
We consider four-dimensional homogeneous pseudo-Riemannian manifolds with non-trivial isotropy and completely classify the cases giving rise to non-trivial homogeneous Ricci solitons. In particular, we show the existence of non-compact homogeneous (and also invariant) pseudo-Riemannian Ricci solitons which are not isom…
Study finds conditions for austere orbits in pseudo-Riemannian symmetric spaces.
problem Characterizing austere orbits in pseudo-Riemannian symmetric spaces.
method Using restricted root system theory and Cartan subspaces, the study provides an equivalent condition for austere orbits.
result Examples of austere orbits are given.
We describe, by their holonomy groups, all simply connected irreducible non-locally symmetric pseudo-Riemannian SpinC manifolds which admit parallel spinors. So we generalise the Riemannian case and the pseudo-Riemannian one.
Defines Witt structures for pseudo-Riemannian manifolds.
problem No specific problem stated; focuses on new structure.
method Introduces a connection adapted to Witt structures.
result Revisits geodesics and symmetric spaces in new context.
Study geodesic orbit property on pseudo-Riemannian H-type nilmanifolds.
problem Characterize geodesic orbit property for pseudo-Riemannian H-type Lie groups.
method Extend results from Riemannian to pseudo-Riemannian H-type Lie groups, focusing on minimal admissible Clifford modules.
result Complete characterization of geodesic orbit property for pseudo-Riemannian H-type Lie groups.
New spaces identified with specific properties.
problem Characterizing homogeneous spaces with quaternionic structures.
method Analyzing pseudo-Riemannian almost quaternionic homogeneous spaces with irreducible isotropy.
result Spaces are locally isometric to quaternionic Kähler symmetric spaces under certain conditions.
Invariants from contact structures lead to Einstein-Weyl geometry.
problem Understanding local geometry of contact sub-pseudo-Riemannian structures.
method Constructing invariants and showing conditions for Einstein-Weyl geometry.
result Contact structures on contact manifolds lead to Einstein-Weyl geometries in dimension 3.
The paper classifies helix curves on a pseudo-Riemannian surface.
problem Classifying helix curves on pseudo-Riemannian surfaces.
method Analyzing geodesic flow vector fields and pseudo-Riemannian metrics.
result All helix curves are circular helixes with constant curvature and torsion.
We consider four dimensional conformally flat homogeneous pseudo Riemannian manifolds. According to forms (Seger types) of the Ricci operator, we provide a full classification of four dimensional pseudo Riemannian conformally flat homogeneous Ricci solitons.
We describe, by their holonomy groups, all complete simply connected irreducible non-locally symmetric pseudo-Riemannian SpinC manifolds which admit parallel spinors. So we generalize the Riemannian SpinC case and the pseudo-Riemannian Spin one.
Study pseudo-Riemannian metrics on Jordan superalgebras.
problem No specific problem stated; focus on metrics.
method Coadjoint orbit-like construction for pseudo-Euclidean Jordan superalgebras.
result Investigated canonical pseudo-Riemannian metrics.
The paper studies metrics on Lie groups and their curvatures.
problem Understanding left-invariant pseudo-Riemannian metrics on Lie groups.
method Formulation of a procedure based on moduli space for left-invariant pseudo-Riemannian metrics.
result Left-invariant pseudo-Riemannian metrics on Lie groups of real hyperbolic spaces have constant sectional curvatures.
Defines compatibility between Jacobi structures and pseudo-Riemannian metrics on Jacobi algebroids.
problem Generalizing compatibility between Poisson and pseudo-Riemannian metrics to Jacobi structures.
method Introduces and studies compatibility conditions for Jacobi structures and pseudo-Riemannian metrics on Jacobi algebroids.
result Compatibility conditions are preserved under Poissonization and equivalent to Sasakian structures for contact pseudo-metrics.
Classifies semisimple weakly symmetric pseudo-Riemannian manifolds.
problem Classifying pseudo-Riemannian manifolds with specific properties.
method Developed from compact Lie group cases, analyzed isotropy representation and metric signature.
result Obtained classification of semisimple weakly symmetric manifolds of specific signatures.