Distinguished Vector Fields Research Articles

On the Potential Vector Fields of Soliton-Type Equations

We highlight some properties of a class of distinguished vector fields associated to a (1,1)-tensor field and to an affine connection on a Riemannian manifold, with a special view towards the Ricci vector fields, and we characterize them with respect to statistical, almost Kähler, and locally product structures. In particular, we provide conditions for these vector fields to be closed, Killing, parallel, or semi-torse forming. In the gradient case, we give a characterization of the Euclidean sphere. Among these vector fields, the Ricci and torse-forming-like vector fields are particular cases.

Isometries and the double copy

In the standard derivation of the Kerr-Schild double copy, the geodicity of the Kerr-Schild vector and the stationarity of the spacetime are presented as assumptions that are necessary for the single copy to satisfy Maxwell’s equations. However, it is well known that the vacuum Einstein equations imply that the Kerr-Schild vector is geodesic and shear-free, and that the spacetime possesses a distinguished vector field that is simultaneously a Killing vector of the full spacetime and the flat background, but need not be timelike with respect to the background metric. We show that the gauge field obtained by contracting this distinguished Killing vector with the Kerr-Schild graviton solves the vacuum Maxwell equations, and that this definition of the Kerr-Schild double copy implies the Weyl double copy when the spacetime is Petrov type D. When the Killing vector is taken to be timelike with respect to the background metric, we recover the familiar Kerr-Schild double copy, but the prescription is well defined for any vacuum Kerr-Schild spacetime and we present new examples where the Killing vector is null or spacelike. While most examples of physical interest are type D, vacuum Kerr-Schild spacetimes are generically of Petrov type II. We present a straightforward example of such a spacetime and study its double copy structure. Our results apply to real Lorentzian spacetimes as well as complex spacetimes and real spacetimes with Kleinian signature, and provide a simple correspondence between real and self-dual vacuum Kerr-Schild spacetimes. This correspondence allows us to study the double copy structure of a self-dual analog of the Kerr spacetime. We provide evidence that this spacetime may be diffeomorphic to the self-dual Taub-NUT solution.

Geodesic and Newtonian Vector Fields and Symmetries of Mechanical Systems

Geodesic vector fields and other distinguished vector fields on a Riemann manifold were used in the study of free motions on such a manifold, and we applied the geometric Hamilton–Jacobi theory for the search of geodesic vector fields from Hamilton–Jacobi vector fields and the same for closed vector fields. These properties were appropriately extended to the framework of Newtonian and generalised Newtonian systems, in particular systems defined by Lagrangians of the mechanical type and velocity-dependent forces. Conserved quantities and a generalised concept of symmetry were developed, particularly for Killing vector fields. Nonholonomic constrained Newtonian systems were also analysed from this perspective, as well as the relation among Newtonian vector fields and Hamilton–Jacobi equations for conformally related metrics.

Nonsingular extension of the Kerr-NUT–(anti–)de Sitter spacetimes

In 1963 Ezra Ted Newman and his two students Louis A. Tamburino, and Theodore W. J. Unti, proposed a deformation of the Shwarzschild spacetime that made it twisting. In the cosmological context, an equivalent solution had been found earlier, in 1951, by Abraham Haskel Taub. The problem that these solutions have is a conical singularity along the symmetry axis at all distances from the origin. In 1969 Misner proposed a non-singular interpretation of Taub-NUT spacetimes. We extend and refine his method to include a broader family of solutions and completely solve the outstanding issue of a non-singular extension of the Kerr-NUT- (anti) de Sitter solutions to Einstein's equations. Our approach relies on an observation that in 2 dimensional algebra of Killing vector fields there exist 2 distinguished vector fields that may be used to define $U(1)$-principal bundle structure over the non-singular spaces of non-null orbits. For all admissible parameters we derive appropriate Killing vector fields and discuss limits to spacetimes with less parameters. The global structure of spacetime, together with non-singular conformal geometry of the infinities is presented and (possibly also projectively non-singular) Killing horizons is presented.

Some classes of CR submanifolds with an umbilical section of the nearly Kähler [formula omitted

Recently, the investigation of a CR submanifolds of the nearly Kähler manifold S3×S3 was started. In this paper it is proved that CR submanifolds of the nearly Kähler manifold S3×S3 with umbilical sections must have dimension three and then we obtain some examples of them with distinguished vector fields. Also, we classify minimal submanifolds that have a vector field E4 as an umbilical section. The main result is classification of the three-dimensional umbilical CR submanifolds with totally geodesic and almost complex distribution D1.

Canonical Kähler metrics on classes of Lorentzian 4-manifolds

Conditions for the existence of Kahler–Einstein metrics and central Kahler metrics (Maschler in Trans Am Math Soc 355:2161–2182, 2003) along with examples, both old and new, are given on classes of Lorentzian 4-manifolds with two distinguished vector fields. The results utilize the general construction (Aazami and Maschler in Kahler metrics via Lorentzian geometry in dimension four, Complex Manifolds 7:36–61 (2020) of Kahler metrics on such manifolds. The examples include both complete and incomplete metrics, and some reside on Lie groups associated with four types of Lie algebras. An appendix includes a similar construction for scalar-flat Kahler metrics.

Kähler metrics via Lorentzian Geometry in dimension four

AbstractGiven a semi-Riemannian 4-manifold (M, g) with two distinguished vector fields satisfying properties determined by their shear, twist and various Lie bracket relations, a family of Kähler metrics gK is constructed, defined on an open set in M, which coincides with M in many typical examples. Under certain conditions g and gK share various properties, such as a Killing vector field or a vector field with a geodesic flow. In some cases the Kähler metrics are complete. The Ricci and scalar curvatures of gK are computed under certain assumptions in terms of data associated to g. Many examples are described, including classical spacetimes in warped products, for instance de Sitter spacetime, as well as gravitational plane waves, metrics of Petrov type D such as Kerr and NUT metrics, and metrics for which gK is an SKR metric. For the latter an inverse ansatz is described, constructing g from the SKR metric.

Structure theory for ensemble controllability, observability, and duality

Ensemble control deals with the problem of using a finite number of control inputs to simultaneously steer a large population (in the limit, a continuum) of control systems. Dual to the ensemble control problem, ensemble estimation deals with the problem of using a finite number of measurement outputs to estimate the initial state of every individual system in the ensemble. We introduce in the paper a novel class of ensemble systems, termed distinguished ensemble systems, and establish sufficient conditions for controllability and observability of such systems. Every distinguished ensemble system has two key components, namely a set of distinguished control vector fields and a set of codistinguished observation functions. Roughly speaking, a set of vector fields is distinguished if it is closed (up to scaling) under Lie bracket, and moreover, every vector field in the set can be obtained by a Lie bracket of two vector fields in the same set. Similarly, a set of functions is codistinguished to a set of vector fields if the Lie derivatives of the functions along the given vector fields yield (up to scaling) the same set of functions. We demonstrate in the paper that the structure of a distinguished ensemble system can significantly simplify the analysis of ensemble controllability and observability. Moreover, such a structure can be used as a guiding principle for ensemble system design. We further address in the paper the problem about existence of a distinguished ensemble system for a given manifold. We provide an affirmative answer for the case where the manifold is a connected semi-simple Lie group. Specifically, we show that every such Lie group admits a set of distinguished vector fields, together with a set of codistinguished functions. The proof is constructive, leveraging the structure theory of semi-simple real Lie algebras and representation theory. Examples will be provided along the presentation of the paper illustrating key definitions and main results.

Mean Curvature Flow Solitons in the Presence of Conformal Vector Fields

In this paper we introduce and study a notion of mean curvature flow soliton in Riemannian ambient spaces general enough to encompass target spaces of constant sectional curvature, Riemannian products or, in increasing generality, warped product spaces. As expected, our definition is motivated by the self-similarity of certain special solutions of the mean curvature flow with respect to the flow generated by a distinguished vector field on the target manifold. Our approach allows us to identify some natural geometric quantities that satisfy elliptic equations or differential inequalities in a simple and manageable form for which the machinery of weak maximum principles is valid. The latter is one of the main tools we apply to derive several new characterizations and rigidity results for mean curvature flow solitons that extend to our much more general setting known properties, for instance, in Euclidean space.

Geometry of Control-Affine Systems

Motivated by control-affine systems in optimal control theory, we introduce the notion of a point-ane distribution on a manifold X - i.e., an affine distribution F together with a distinguished vector field contained in F. We compute local invariants for point-affine distributions of constant type when dim(X) = n, rank(F) = n 1, and when dim(X) = 3, rank(F) = 1. Unlike linear distributions, which are characterized by integer- valued invariants - namely, the rank and growth vector - when dim(X) 4, we find local invariants depending on arbitrary functions even for rank 1 point-affine distributions on manifolds of dimension 2.

On a class of Einstein hypersurfaces immersed in a Riemannian manifold

Let x:M→\(\) be an isometric immersion of a hypersurface M into an (n+1)-dimensional Riemannian manifold \(\) and let ρ i (i∈{1,...,n}) be the principal curvatures of M. We denote by E and P the distinguished vector field and the curvature vector field of M, respectively, in the sense of [8].¶If M is structured by a P-parallel connection [7], then it is Einsteinian. In this case, all the curvature 2-forms are exact and other properties induced by E and P are stated.¶The principal curvatures ρ i are isoparametric functions and the set (ρ1,...,ρ n ) defines an isoparametric system [10].¶In the last section, we assume that, in addition, M is endowed with an almost symplectic structure. Then, the dual 1-form π=P♭ of P is symplectic harmonic. If M is compact, then its 2nd Betti number b2≥1.

On Contact Submanifolds in Complex Projective Spaces

We treat n-dimensional real submanifolds of complex projective spaces in the case when the maximal holomorphic tangent subspace is (n - 1)-dimensional. In particular, we study the case when the induced almost contact structure on a submanifold is contact, we establish a few characteristics of the shape operator with respect to the distinguished vector field and we give one characterization of a contact submanifold in this case

CR submanifolds of maximal CR dimension of complex projective space

We treat CR submanifolds of the complex projective space with the maximal CR dimension, we establish a few characteristics of the shape operator with respect to the distinguished vector field and we give one characterization of the CR submanifold in this particular case.

On the 1953 Barthel connection of a Finsler space and its mathematical and physical interpretation

A precise formulation of the Barthel connection in Finsler spaces is given, based on the general theory of connections in fibre bundles. It is shown that, if a nonzero vector field Y is given, the Barthel connection is closely related to the Cartan connection and that then an adjoint Riemannian space can be defined and the Barthel connection is metrical in the Riemannian space. A short discussion of the Finsler spaces with (α,β)-metric is given and the problem of physical interpretation is briefly remarked. It seems, in the light of the result of our paper, that the Barthel connection can be considered as the connection of a Finsler space with a distinguished vector field, similarly as the Levi-Civita connection is the connection of a Riemannian space.

Relativistic and non-relativistic classical field theory on five-dimensional spacetime

This paper is a sequel to earlier ones, (1984), in which, on the one hand, classical field theories were described on a curved Newtonian spacetime, and, on the other hand, the Newtonian gravitation theory was formulated on a five-dimensional spacetime with a metric of signature (++++-) and a covariantly constant vector field. Here the authors show that Lagrangians for matter fields are easily formulated on this extended spacetime from simple invariance arguments and that stress-energy tensors can be derived from them in the usual manner so that four-dimensional spacetime expressions are obtained that are consistent in the relativistic as well as the Newtonian case. In the former the theory is equivalent to general relativity. When the magnitude of the distinguished vector field vanishes equations for the (covariant) Newtonian limit follow. They demonstrate this here explicitly in the case of the Klein-Gordon/Schrodinger and the Dirac field and its covariant non-relativistic analogue, the Levy-Leblond field. In especially the latter example the covariant Newtonian theory simplifies dramatically in this five-dimensional form.