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 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 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.
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.
The study classifies quasi-minimal surfaces in 4D pseudo-Riemannian space-forms with positive nullity.
problem Characterizing surfaces in pseudo-Riemannian space forms with positive nullity.
method Analyzing the relative null space and classifying quasi-minimal surfaces.
result Classifications of quasi-minimal surfaces with positive relative nullity.
Two pseudo-Riemannian metrics g and gˉ are geodesically equivalent, if they share the same (unparameterized) geodesics. We give a complete local description of such metrics which solves the natural generalisation of Beltrami problem for pseudo-Riemannian metrics.
We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces, prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces, and prove t…
Study finds conditions for existence of specific pseudo-Riemannian cobordisms.
problem Existence of Spin(n+1)-dimensional cobordisms with specific signature. method Analyzes Spin(n+1)-dimensional cobordisms with signature (2,n−1). result Computes cobordism groups and necessary/sufficient conditions for existence.
The paper classifies submanifolds in pseudo-Riemannian space forms.
problem Characterizing and classifying totally umbilical submanifolds.
method Classification of congruent classes of totally umbilical submanifolds in non-flat pseudo-Riemannian space forms.
result Some moduli spaces of isometric immersions between space forms are non-Hausdorff.
Study of pseudo-Riemannian metrics related to Monge-Ampère structures.
problem Properties of pseudo-Riemannian metrics and Hessian structures.
method Analysis of Monge-Ampère structures and Ricci flat solutions.
result Description of a family of Ricci flat solutions parametrized by six coefficients.
Generalized tensor analysis in the sense of Colombeau's construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions we define the notions of generalized pseudo-Riemannian met…
Study on Einstein manifolds with specific properties.
problem Identifying all locally homogeneous compact pseudo-Riemannian Einstein manifolds.
method Analyzing standard compact Clifford-Klein forms of simple non-compact Lie groups and conjecturing based on T. Kobayashi's work.
result Found at least one Einstein metric in standard compact Clifford-Klein forms and conjecturing these are the only possible ones.
Geometric calculus introduced on pseudo-Riemannian manifolds without embedding.
problem Developing calculus on pseudo-Riemannian manifolds without embedding.
method Direct axiomatic approach to geometric calculus, paralleling general relativity.
result Full theory of differential calculus for vector, multivector, and tensor fields developed.
Study real GIT and Wick-rotations of pseudo-Riemannian manifolds.
problem Understanding Wick-rotations and their conditions for pseudo-Riemannian manifolds.
method Extending earlier results, providing sufficient and necessary conditions for Wick-rotatability, and deriving an invariance theorem.
result Derived sufficient and necessary conditions for pseudo-Riemannian manifolds to be Wick-rotatable.
Curvature measures uniquely determined by invariance under embeddings.
problem Characterizing curvature measures uniquely.
method Applied Weyl principle and Künneth-type formula.
result Curvature measures uniquely characterized by invariance under isometric embeddings.
We construct local normal forms of pseudo-Riemannian projectively equivalent 2-dimensional metrics.
The paper extends graph embedding models to handle multiple relations.
problem Link prediction in multi-relational networks.
method Generalized pseudo-Riemannian embedding models to multi-relational networks, considering relations as submanifolds.
result Validation of the approach in link prediction tasks, including knowledge graph completion and biological domain analysis.
Study pseudo-Riemannian Sasaki metrics on solvable Lie groups.
problem Characterize and classify pseudo-Riemannian Sasaki solvmanifolds.
method Sasaki reduction and pseudo-Kähler quotient under Reeb vector field action.
result Classify pseudo-Riemannian Sasaki solvmanifolds in dimensions 5 and 7.
There is a well developed theory of weakly symmetric Riemannian manifolds. Here it is shown that several results in the Riemannian case are also valid for weakly symmetric pseudo-Riemannian manifolds, but some require additional hypotheses. The topics discussed are homogeneity, geodesic completeness, the geodesic orbit…
Classification theorems for Ricci solitons on Minkowski hypersurfaces.
problem Characterizing Ricci solitons on pseudo-Riemannian hypersurfaces in Minkowski space.
method Analyzing the shape operator and potential vector field of hypersurfaces.
result Characterization of Ricci solitons on various types of hypersurfaces in Minkowski space.
We provide examples of naturally reductive pseudo-Riemannian spaces, in particular an example of a naturally reductive pseudo-Riemannian 2-step nilpotent Lie group (N,<,>N), such that <,>N is invariant under a left action and for which the center is degenerate. The metric does not correspond to a bi-in…
Wick rotations between pseudo-Riemannian spaces using real GIT.
problem Existence and non-existence of Wick-rotations between different pseudo-Riemannian spaces.
method Using real GIT to study structure groups of pseudo-Riemannian spaces as real forms of complex Lie groups.
result Derivation of new results regarding the existence and non-existence of Wick-rotations.
New pseudo-Riemannian nilmanifolds classified from earlier work.
problem Classifying weakly symmetric pseudo-Riemannian nilmanifolds.
method From earlier classification of weakly symmetric pseudo-Riemannian manifolds, extended to nilmanifolds with compact H and nilpotent N. result A plethora of new isometric pseudo-Riemannian nilmanifolds identified.