arXiv research

Introduces Kundt spaces using geometric language.

Classifies left invariant Kundt structures on 3D Lie groups.

In this paper we consider a new approach to studying Kundt spacetimes through $G$-structures. We define a Lie-group $GN$ such that the $GN$-structures satisfying an integrability condition and an existence criterion, which we call Kundt structures, have the property that each metric belonging to the Kundt structure is …

Researchers found differential invariants for Kundt spacetimes.

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.

Study isotropic solutions in smooth metric measure spaces with vacuum Einstein equations.

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.

This paper is concerned with deformations of Kundt metrics in the direction of type $III$ tensors and nil-Killing vector fields whose flows give rise to such deformations. We find various characterizations within the Kundt class in terms of nil-Killing vector fields and obtain a theorem classifying algebraic stability …

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.

Study of four-dimensional Lorentzian manifolds with real Killing spinors.

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.

Study finds holonomy algebras for Lorentzian Weyl spin manifolds with specific spinors.

Study of differential spinors on three-manifolds with skew-torsion.

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 ${\bf D}$ relative to some common null frame. Such spacetimes are known as type ${\b…

Complete description of flat Lorentzian Lie groups solved.

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 ${\mathbb R}^2$ 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 $\mathcal{I}$-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 $(α,β)$-Finsler spaces, respectively $(A,B)$-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…

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 $2$-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.

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.

(1,1) non-L-space knots are foliar in 3D space.

Universal spaces for finite topological spaces simplify shape descriptions.

The paper extends Stone duality to topological convexity spaces.

No Einstein hypersurfaces found in Damek-Ricci spaces.

The distance function $\varrho(p,q)$ (or $d(p,q)$) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over $\mathbb R^n$, 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.

Metric spaces uniquely split into Hilbert and non-line-split parts.

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.

New curvature positivity helps classify spherical spaces and complex projective spaces.

Study of complete space-like self-expanders in Minkovski space.

The abstract discusses the linear and smooth structures of mapping spaces.

Study on convergence of transformed metric spaces as dimensions grow.

Characterizes when almost smooth spaces become RCD spaces.

Stability of Wasserstein spaces under various convergence types.

New infinite families of flat spaces found from symmetric spaces.

New quasi space forms solve Thurston's geometrical space form problem.