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. 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 paper classifies conformal solitons in pseudo-Euclidean spaces.
problem Classifying conformal solitons in pseudo-Euclidean spaces.
method Classification through pseudo-Riemannian hypersurfaces and position vector fields.
result Complete classification of conformal solitons in pseudo-Euclidean spaces.
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.
The paper examines 4D hypersurfaces with constant mean curvature in pseudo-Riemannian space forms.
problem Investigating properties of 4D hypersurfaces with specific curvature conditions.
method Analyzing hypersurfaces with proper mean curvature vector field in pseudo-Riemannian space forms.
result Bi-harmonic hypersurfaces in N^5_s(c) are minimal in certain cases.
Study of conformal mappings in pseudo-Finsler spaces, extending results from pseudo-Riemannian geometry.
problem Understanding conformal mappings in pseudo-Finsler spaces and their properties.
method Extending results from pseudo-Riemannian geometry and proposing a technique to reduce problems involving pseudo-Finslerian conformal vector fields.
result Conformal groups of flat pseudo-Finsler spaces can be richer than flat Finslerian and pseudo-Euclidean conformal groups.
Defines virtual immersions to characterize symmetric spaces.
problem Characterizing symmetric spaces using virtual immersions.
method Defines virtual immersions as a generalization of isometric immersions, proves characterization of symmetric spaces.
result A manifold admits a virtual immersion with skew symmetric second fundamental form if and only if it is a symmetric space.
Conformally quasi-recurrent (CQR)_n pseudo-Riemannian manifolds are investigated, and several new results are obtained. It is shown that the Ricci tensor and the gradient of the fundamental vector are Weyl compatible tensors (the notion was introduced recently by the authors and applies to significative space-times), (…
Study geodesic mappings and concircular fields in pseudo-Riemannian manifolds.
problem Characterize geodesic mappings and concircular fields in pseudo-Riemannian manifolds.
method Analyze geodesic mappings and concircular fields on Vn(K)-spaces, proving properties of the set of solutions. result The set of solutions of geodesic mappings on Vn(K)-spaces forms a special Jordan algebra, and the set of solutions generated by consircular fields is an ideal of this algebra. The theory of harmonic vector fields on Riemannian manifolds is generalised to pseudo-Riemannian manifolds. Harmonic conformal gradient fields on pseudo-Euclidean hyperquadrics are classified up to congruence, as are harmonic Killing fields on pseudo-Riemannian quadrics. A para-Kaehler twisted anti-isometry is used to …
The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denot…
New unitary representations constructed for complex nilmanifolds.
problem Constructing unitary representations on specific geometric spaces.
method Combining geometric methods with recent developments on pseudo-Riemannian nilmanifolds.
result Developed theory for square integrable representations on complex nilmanifolds.
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…
Study on conformal vector fields on Lie groups, proving properties and providing examples.
problem Characterizing conformal vector fields on Lie groups with pseudo-Riemannian metrics.
method Investigation of left-invariant conformal vector fields on Lie groups with specific metrics, proving properties and providing examples.
result Necessary conditions and examples of non-Killing conformal vector fields on non-unimodular Lie groups.
Invariant covariant derivatives on homogeneous spaces are characterized.
problem Understanding invariant covariant derivatives on homogeneous spaces.
method Expressing covariant derivatives in terms of horizontally lifted vector fields and bilinear maps.
result Existence and characterization of invariant covariant derivatives.
We study the isometry groups and Killing vector fields of a family of pseudo-Riemannian metrics on Euclidean space which have neutral signature (3+2p,3+2p). All are p+2 curvature homogeneous, all have vanishing Weyl scalar invariants, all are geodesically complete, and all are 0-curvature modeled on an indecomposible s…
In this paper we develop the basic tools for a classification of Killing vector fields of constant length on pseudo--riemannian homogeneous spaces. This extends a recent paper of M. Xu and J. A. Wolf, which classified the pairs (M,ξ) where M=G/H is a Riemannian normal homogeneous space, G is a compact simple Li…
We describe a new class of holonomy groups on pseudo-Riemannian manifolds. Namely, we prove the following theorem. Let g be a nondegenerate bilinear form on a vector space V, and L:V -> V a g-symmetric operator. Then the identity component of the centraliser of L in SO(g) is a holonomy group for a suitable Levi-Civita …
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 on biharmonic and interpolating sesqui-harmonic vector fields on para-Kähler--Norden manifolds.
problem Investigating higher-order harmonicity in pseudo-Riemannian geometry.
method Deriving first variations of bienergy and interpolating sesqui-energy functionals, characterizing biharmonic and interpolating sesqui-harmonic vector fields.
result Explicit characterizations and examples of vector fields satisfying biharmonic and interpolating sesqui-harmonic conditions.
Improving a result of Eschenburg and Kim we give a criterion for semisimplicity of pseudo-Riemannian extrinsic symmetric spaces in terms of the shape operator with respect to the mean curvature vector.
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.
A submanifold of a pseudo-Riemannian manifold is said to have parallel mean curvature vector if the mean curvature vector field H is parallel as a section of the normal bundle. Submanifolds with parallel mean curvature vector are important since they are critical points of some natural functionals. In this paper, we su…
Study on Ricci solitons on tangent and unit tangent bundles.
problem Characterizing Ricci solitons on tangent and unit tangent bundles.
method Analyzing pseudo-Riemannian g-natural metrics and their Ricci soliton properties. result Classification of conformal vector fields and existence of non-Einstein Ricci solitons.
We study the geometry of pseudo-Riemannian manifolds which are Jacobi--Tsankov, i.e. J(x)J(y)=J(y)J(x) for all tangent vectors x and y. We also study manifolds which are 2-step Jacobi nilpotent, i.e. J(x)J(y)=0 for all tangent vectors x and y.
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…
Let M be a space-like surface immersed in a 4-dimensional pseudo-Riemannian space form R24(c) with constant sectional curvature c and index two. In the first part of this article, we prove that the Gauss curvature K, the normal curvature KD, and mean curvature vector H of M satisfy the general inequali…
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.
The paper studies holomorphic curves in a pseudo-Riemannian space and their moduli space.
problem Understanding the moduli space of holomorphic curves in a pseudo-Riemannian space.
method Using Frenet framing and G2′-Higgs bundles, the paper describes the moduli space of equivariant alternating holomorphic curves. result Equivariant alternating holomorphic curves are infinitesimally rigid.
We discuss Lie-Tresse theorem for the pseudogroup of diffeomorphisms acting on the space of (pseudo-)Riemannian metrics, and relate this to existence of Killing vector fields. Then we discuss the impact of symmetry in the general case.
Enhances Hamiltonian systems stability through generalized double bracket vector fields.
problem Stabilizing already stable points in Hamiltonian systems.
method Generalized double bracket vector fields on Poisson manifolds with pseudo-Riemannian metrics.
result Enhanced equilibria stability through dissipation terms.
I prove the two-dimensional pseudo-Riemannian version of the projective Obata conjecture stating that on a closed manifold different from the round sphere every projective (i.e., geodesic-preserving) vector field is Killing.
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.
Researchers classify differential operators for anti-de Sitter spaces.
problem Classifying differential operators intertwining representations of conformal vector fields.
method Constructing and classifying differential operators between spaces of differential forms.
result Classification of differential operators for anti-de Sitter spaces.
Geodesic completeness proven for certain symmetric spaces.
problem Geodesic completeness of compact locally symmetric pseudo-Riemannian manifolds of signature (2,2).
method Analysis of unique parallel lightlike vector fields and group actions.
result Geodesic completeness established for compact locally symmetric spaces modeled on specific spaces.
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.
This is the first of two companion papers in which 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 proper…
Develops geometric Weyl calculus for curved spacetimes.
problem Quantum field theory on curved spacetimes.
method Geometric framework for Weyl quantization on pseudo-Riemannian manifolds.
result Explicit computations of Weyl symbols for various operators.
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…
We establish normal forms for conformal vector fields on pseudo-Riemannian manifolds in the neighborhood of a singularity. For real-analytic Lorentzian manifolds, we show that the vector field is analytically linearizable or the manifold is conformally flat. In either case, the vector field is locally conjugate to a no…
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.
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.
This research bridges Killing vectors and Lie algebras through induced vector fields.
problem Understanding the relationship between Killing vector fields and Lie algebras.
method Defining and exploring induced vector fields to connect Killing vector fields with isometry Lie groups.
result Established a new connection between Killing vector fields and Lie algebras.
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.
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.
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.
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.
We investigate Kaluza-Klein metrics with a recurrent light-like vector field over a pseudo-Riemannian manifold.