Introduction to Kundt spacesmath.DG
Introduces Kundt spaces using geometric language.
problem Clarify properties of Kundt spaces.
method Geometric approach focusing on key concepts.
result Pedagogical introduction to Kundt spaces.
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48 results for “Kundt spaces”
Classifies left invariant Kundt structures on 3D Lie groups.
problem Understanding Kundt spacetimes and their properties.
method Analyzes local structure and properties of left invariant Kundt structures.
result Classifies all left invariant Kundt structures on 3D simply connected unimodular Lie groups.
Kundt Structuresmath.DG
New approach to Kundt spacetimes using -structures.
problem Understanding Kundt spacetimes through -structures.
method Define Lie-group and Kundt structures with integrability and existence criteria.
result Characterization of all left invariant Kundt structures on homogeneous manifolds.
Researchers found differential invariants for Kundt spacetimes.
problem Equivalence problem for Lorentzian metrics in Kundt spacetimes.
method Found generators for rational differential invariants for Kundt spacetimes.
result Relating findings to other approaches to the equivalence problem.
The paper studies deformations of Kundt metrics using nil-Killing vector fields.
problem Deformations of Kundt metrics in the direction of type III tensors.
method Characterizations within the Kundt class using nil-Killing vector fields.
result Theorem classifying algebraic stability of tensors and sufficient criteria for preserving spi's.
Vacuum Kundt Wavesgr-qc
We discuss the invariant classification of vacuum Kundt waves using the Cartan-Karlhede algorithm, and the upper bound on the number of iterations of the Karlhede algorithm to classify the vacuum Kundt waves. By choosing a particular coordinate system we partially construct the canonical coframe used in the classificat…
Characterizes spacetimes using doubly torqued vectors.
problem Classifying spacetimes based on their geometric properties.
method Characterization through doubly torqued vectors and their properties.
result Doubly twisted and Kundt spacetimes can be characterized.
Study isotropic solutions in smooth metric measure spaces with vacuum Einstein equations.
problem Investigate isotropic solutions in smooth metric measure spaces under vacuum Einstein field equations.
method Define a weighted Einstein tensor and associated vacuum field equations. Analyze solutions for different spacetime types.
result Isotropic solutions have nilpotent Ricci operator and specific forms in 2- and 3-step nilpotent manifolds.
A pseudo-Riemannian manifold is called CSI if all scalar polynomial invariants constructed from the curvature tensor and its covariant derivatives are constant. In the Lorentzian case, the CSI spacetimes have been studied extensively due to their application to gravity theories. It is conjectured that a CSI spacetime i…
Proves the Kundt conjecture in arbitrary dimensions, confirming its validity.
problem Determining spacetimes not characterized by scalar polynomial curvature invariants.
method New bilinear map and analysis of covariant derivatives of the Riemann tensor.
result Confirms the Kundt conjecture in arbitrary dimensions, removing regularity assumptions.
Kundt waves belong to the class of spacetimes which are not distinguished by their scalar curvature invariants. We address the equivalence problem for the metrics in this class via scalar differential invariants with respect to the equivalence pseudo-group of the problem. We compute and finitely represent the algebra o…
It is of interest to study supergravity solutions preserving a non-minimal fraction of supersymmetries. A necessary condition for supersymmetry to be preserved is that the spacetime admits a Killing spinor and hence a null or timelike Killing vector field. Any spacetime admitting a covariantly constant null vector fiel…
Impulsive waves contradict a 1962 conjecture about pp-waves.
problem The failure of the Ehlers--Kundt conjecture in the impulsive case.
method Summarized completeness results for impulsive wave spacetimes.
result Impulsive pp-waves are complete, contradicting the conjecture.
Study of four-dimensional Lorentzian manifolds with real Killing spinors.
problem Characterizing and understanding four-dimensional Lorentzian manifolds with Killing spinors.
method Differential geometry and topology, Killing spinor equations, flow equations.
result Proves that the evolution flow defined by a real Killing spinor preserves the Hamiltonian and momentum constraints of the Einstein equation with negative curvature.
We study the existence of a non-spacelike isometry, ζ, in higher dimensional Kundt spacetimes with constant scalar curvature invariants (CSI). We present the particular forms for the null or timelike Killing vectors and a set of constraints for the metric functions in each case. Within the class of N dimensional CSI Ku…
We consider restrictions placed by geodesic completeness on spacetimes possessing a null parallel vector field, the so-called Brinkmann spacetimes. This class of spacetimes includes important idealized gravitational wave models in General Relativity, namely the plane-fronted waves with parallel rays, or pp-waves, which…
Study parallel waves in spacetimes, focusing on causality and open questions.
problem Addressing open questions in the field of parallel waves in spacetimes.
method Review and summarize existing results, introduce new concepts like null coordinates and Penrose limits.
result Progress made on the Ehlers-Kundt conjecture.
Study finds holonomy algebras for Lorentzian Weyl spin manifolds with specific spinors.
problem Characterizing Lorentzian Weyl spin manifolds with weighted parallel spinors.
method Analyzing holonomy algebras and introducing special coordinates.
result Local forms and examples of Lorentzian Weyl spin manifolds with weighted parallel spinors.
Study of differential spinors on three-manifolds with skew-torsion.
problem Characterizing differential spinors on Lorentzian three-manifolds with skew-torsion.
method Developed spinorial polyforms and used them to study differential spinors, proving that every differential spinor is equivalent to an isotropic line preserved by a metric connection with skew-torsion.
result Obtained structural results about Lorentzian three-manifolds equipped with skew-torsion parallel spinors, which are necessarily Kundt and geodesically complete in the compact case.
We consider the class of locally boost isotropic spacetimes in arbitrary dimension. For any spacetime with boost isotropy, the corresponding curvature tensor and all of its covariant derivatives must be simultaneously of alignment type relative to some common null frame. Such spacetimes are known as type ${\b…
Complete description of flat Lorentzian Lie groups solved.
problem Long-standing open problem in pseudo-Riemannian Lie groups.
method Refined analysis of double extension process and explicit classification of Lie algebras.
result All flat Lorentzian Lie algebras arise from flat Euclidean ones.
Ehlers-Kundt conjecture is a physical assertion about the fundamental role of plane waves for the description of gravitational waves. Mathematically, it becomes equivalent to a problem on the Euclidean plane with a very simple formulation in Classical Mechanics: given a non-necessarily autonomous potent…
We consider weighted parallel spinors in Lorentzian Weyl geometry in arbitrary dimensions, choosing the weight such that the integrability condition for the existence of such a spinor, implies the geometry to be Einstein-Weyl. We then use techniques developed for the classification of supersymmetric solutions to superg…
We prove a Goldberg-Sachs theorem in dimension three. To be precise, given a three-dimensional Lorentzian manifold satisfying the topological massive gravity equations, we provide necessary and sufficient conditions on the tracefree Ricci tensor for the existence of a null line distribution whose orthogonal complement …
We examine the existence of one parameter groups of diffeomorphisms whose infinitesimal generators annihilate all scalar polynomial curvature invariants through the application of the Lie derivative, known as -preserving diffeomorphisms. Such mappings are a generalization of isometries and appear to be rel…
Berwald geometries are Finsler geometries close to (pseudo)-Riemannian geometries. We establish a simple first order partial differential equation as necessary and sufficient condition, which a given Finsler Lagrangian has to satisfy to be of Berwald type. Applied to -Finsler spaces, respectively -Finsler…
In this paper we consider pseudo-Riemannian spaces of arbitrary signature for which all of the polynomial curvature invariants vanish (VSI spaces). Using an algebraic classification of pseudo-Riemannian spaces in terms of the boost-weight decomposition we first show more generally that a space which is not characterise…
Locally homogeneous pp-wavesmath.DG
We show that every n-dimensional locally homogeneous pp-wave is a plane wave, provided it is indecomposable and its curvature operator, when acting on -forms, has rank greater than one. As a consequence we obtain that indecomposable, Ricci-flat locally homogeneous pp-waves are plane waves. This generalises a classic…
Study investigates metrizability of Finsler spaces with specific metrics.
problem Local metrizability of Finsler spaces with -Kropina metric.
method Investigates the local metrizability of Finsler spaces with -Kropina metric , where is a closed null 1-form.
result Proves that the affine connection on such an -Kropina space is locally metrizable by a (pseudo-)Riemannian metric if and only if the Ricci tensor is symmetric.
While the Lorenzian and Riemanian metrics for which all polynomial scalar curvature invariants vanish (the VSI property) are well-studied, less is known about the four-dimensional neutral signature metrics with the VSI property. Recently it was shown that the neutral signature metrics belong to two distinct subclasses:…
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
(1,1) non-L-space knots are foliar in 3D space.
problem Proving (1,1) non-L-space knots are foliar.
method Analyzing (1,1) non-L-space knots in and lens spaces.
result (1,1) non-L-space knots are persistently foliar.
Universal spaces for finite topological spaces simplify shape descriptions.
problem Describing shape properties of compact metric spaces.
method Inverse limits of finite spaces and Alexandroff extensions.
result Universal spaces simplify shape descriptions of compact metric spaces.
The paper extends Stone duality to topological convexity spaces.
problem Understanding the relationship between topological convexity spaces and sup-lattices.
method Extending Stone duality to topological convexity spaces using preconvexity spaces.
result An adjunction between topological convexity spaces and sup-lattices.
No Einstein hypersurfaces found in Damek-Ricci spaces.
problem Existence of Einstein hypersurfaces in symmetric spaces.
method Analyzing properties of Damek-Ricci spaces and proving no Einstein hypersurface exists.
result No Einstein hypersurfaces in Damek-Ricci spaces.
The distance function (or ) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over , whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no F…
New kernels defined for various spaces, including measures.
problem Defining kernels on non-standard spaces like measures.
method Integrally strictly positive definite and characteristic kernels on Hilbert, Banach, and metric spaces.
result Explicit classes of kernels on spaces and sets of measures.
Metric spaces uniquely split into Hilbert and non-line-split parts.
problem Understanding the structure of metric spaces.
method Proved unique decomposition into Hilbert and non-line-split parts.
result Metric spaces have a unique decomposition into a Hilbert space and a non-line-split part.
In a paper (math.DG/0403528) we obtained explicit examples of Moishezon twistor spaces of some compact self-dual four-manifolds admitting a non-trivial Killing field, and also determined their moduli space. In this note we investigate minitwistor spaces associated to these twistor spaces. We determine their structure, …
Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.
problem Understanding geometric relationships between complex hyperbolic spaces and Hilbert spaces.
method Use a complex analog of the cosine of a vertex angle as a novel technical tool.
result Describe possible triangular faces of tetrahedra in hyperbolic space and three-dimensional subspaces in Hilbert spaces with Pick kernels.
New curvature positivity helps classify spherical spaces and complex projective spaces.
problem Classifying spherical space forms and complex projective spaces.
method Introducing a new positivity notion for curvature.
result Characterizations for spherical space forms and complex projective spaces.
Study of complete space-like self-expanders in Minkovski space.
problem Characterize complete space-like self-expanders in Minkovski space.
method Use of maximum principle of Omori-Yau type to prove rigidity theorems.
result Classification of 2-dimensional complete space-like self-expanders with constant squared norm of the second fundamental form.
The abstract discusses the linear and smooth structures of mapping spaces.
problem The structure of mapping spaces in differential geometry.
method Proving diffeomorphisms and fibre bundle properties.
result Path spaces and base point preserving mapping spaces are Fréchet spaces.
Study on convergence of transformed metric spaces as dimensions grow.
problem Conditions for convergence of transformed metric spaces.
method Clarifying conditions for convergence of transformed spaces from original sequence and vice versa.
result Spheres and projective spaces converge to Gaussian space and its quotient as dimensions increase.
Characterizes when almost smooth spaces become RCD spaces.
problem Understanding conditions for almost smooth spaces to be RCD spaces.
method Characterizations via local volume doubling and Poincaré inequality.
result Characterizes Einstein 4-orbifolds.
Stability of Wasserstein spaces under various convergence types.
problem Stability and finiteness of Wasserstein spaces over singular and non-singular spaces.
method Gromov--Hausdorff convergence and equivariant Gromov--Hausdorff convergence.
result Analogue of Perelman's stability theorem on Wasserstein spaces.
New infinite families of flat spaces found from symmetric spaces.
problem Finding new flat homogeneous spaces.
method Starting from compact symmetric spaces, constructing infinite families of compact homogeneous spaces with invariant Bismut connections.
result Infinite families of compact homogeneous spaces with vanishing Ricci tensor.
New quasi space forms solve Thurston's geometrical space form problem.
problem Solving Thurston's geometrical space form problem.
method Introducing quasi space forms as non-real space forms with specific geometric properties.
result Quasi space forms offer a metrical, local geometrical solution to Thurston's problem.