Narrow Results

We review our recent relativistic generalization of the Gutzwiller–Duistermaat–Guillemin trace formula and Weyl law on globally hyperbolic stationary space-times with compact Cauchy hypersurfaces. We also discuss anticipated generalizations to non-compact Cauchy hypersurface cases.

In the present paper we classify curves and surfaces in Euclidean 3-space which make constant angle with a certain Killing vector field. Moreover, we characterize the catenoid and Dini's surface in terms of constant angle surfaces.

Recently the third author studied horizontal Laplacians in real Finsler vector bundles and complex Finsler manifolds. In this paper, we introduce a class of g-natural metrics G_a,b on the tangent bundle of a Finsler manifold (M, F) which generalizes the associated Sasaki–Matsumoto metric and Miron metric. We obtain the Weitzenböck formula of the horizontal Laplacian associated to G_a,b, which is a second-order differential operator for general forms on tangent bundle. Using the horizontal Laplacian associated to G_a,b, we give some characterizations of certain objects which are geometric interest (e.g. scalar and vector fields which are horizontal covariant constant) on the tangent bundle. Furthermore, Killing vector fields associated to G_a,b are investigated.

We discuss the geodesic connectedness problem in open subsets with convex boundary of globally hyperbolic spacetimes endowed with a complete, timelike or lightlike, Killing vector field. Furthermore, we furnish applications to generalized plane waves.

The object of the present paper is to study a spacetime admitting conharmonic curvature tensor and some geometric properties related to this spacetime. It is shown that in a conharmonically flat spacetime with cyclic parallel Ricci tensor, the energy–momentum tensor is cyclic parallel and conversely. Finally, we prove that for a radiative perfect fluid spacetime if the energy–momentum tensor satisfying the Einstein’s equations without cosmological constant is generalized recurrent, then the fluid has vanishing vorticity and the integral curves of the vector field $U$ are geodesics.

The paper aims to initiate a systematic study of conformal mappings between Finsler spacetimes and, more generally, between pseudo-Finsler spaces. This is done by extending several results in pseudo-Riemannian geometry which are necessary for field-theoretical applications and by proposing a technique that reduces some problems involving pseudo-Finslerian conformal vector fields to their pseudo-Riemannian counterparts. Also, we point out, by constructing classes of examples, that conformal groups of flat (locally Minkowskian) pseudo-Finsler spaces can be much richer than both flat Finslerian and pseudo-Euclidean conformal groups.

If $M$ is a 3-dimensional contact metric manifold such that $Q\phi =\phi Q$ which admits a Yamabe soliton $(g,V)$ with the flow vector field $V$ pointwise collinear with the Reeb vector field $\xi$, then we show that the scalar curvature is constant and the manifold is Sasakian. Moreover, we prove that if $M$ is endowed with a Yamabe soliton $(g,V)$, then either $M$ is flat or it has constant scalar curvature and the flow vector field $V$ is Killing. Furthermore, we show that if $M$ is non-flat, then either $M$ is a Sasakian manifold of constant curvature $1$ or $V$ is an infinitesimal automorphism of the contact metric structure on $M$.

In the present paper, the relation between invariants of the pseudo null curves and the variational vector fields of semi-Riemannian manifolds is introduced. After that, the Killing equations are written in terms of the Bishop curvatures along the pseudo null curve. By means of this approach, Killing equations make allow to interpret the movement of charged particles within the magnetic field. Afterwards, as an application, pseudo null magnetic curves are defined using the Killing variational vector field. The parametric representations of all pseudo null magnetic curves are determined in semi-Riemannian space form. Moreover, various examples of pseudo null magnetic curves are illustrated.

Let $(M,g)$ be a pseudo-Riemannian manifold and $T^{2}M$ be its second-order tangent bundle equipped with the deformed $2$nd lift metric $\overline{g}$ which is obtained from the $2$nd lift metric by deforming the horizontal part with a symmetric $(0,2)$-tensor field $c$. In the present paper, we first compute the Levi-Civita connection and its Riemannian curvature tensor field of $(T^{2}M,\overline{g})$. We give necessary and sufficient conditions for $(T^{2}M,\overline{g})$ to be semi-symmetric. Secondly, we show that $(T^{2}M,\overline{g})$ is a plural-holomorphic $B$-manifold with the natural integrable nilpotent structure. Finally, we get the conditions under which $(T^{2}M,\overline{g})$ with the $2$nd lift of an almost complex structure is an anti-Kähler manifold.

The aim of this paper is to investigate Killing magnetic trajectories of varying electrically charged particles in a three-dimensional warped product $I{\times }_f{𝔼}^2$ with positive warping function $f$, where $I$ is an open interval in $ℝ$ equipped with an induced semi-Euclidean metric on $ℝ$. First, Killing vector fields on $I{\times }_f{𝔼}^2$ are characterized and it is observed that lifts to $I{\times }_f{𝔼}^2$ of Killing vector fields tangent to ${𝔼}^2$ are also Killing on $I{\times }_f{𝔼}^2$. Now, any Killing vector field on $I{\times }_f{𝔼}^2$ corresponds to a Killing magnetic field on $I{\times }_f{𝔼}^2$. Magnetic trajectories (also known as magnetic curves) of charged particles which move under the influence of Lorentz force generated by Killing magnetic fields on $I{\times }_f{𝔼}^2$ are obtained in both Riemannian and Lorentzian cases. Moreover, some examples are exhibited with pictures determining Killing magnetic trajectories in hyperbolic $3$-space ${ℍ}^3(-1)$ modeled by the Riemannian warped product $𝔼{\times }_{exp(z)}{𝔼}^2$. Furthermore, some examples of spacelike, timelike and lightlike Killing magnetic trajectories are given with their possible graphs in the Lorentzian warped product ${𝔼}_1^1{\times }_{exp(-z)}{𝔼}^2$.

In this study, we provide a brief description of rotational surfaces in 4-dimensional (4D) Galilean space using a curve and matrices in $G_4$. That is, we provide different types of rotational matrices, which are the subgroups of $M$ by rotating a selected axis in $E^4$. Hence, we choose two parameter matrices groups of rotations and we give the matrices of rotation corresponding to the appropriate subgroup in Galilean 4-space and we generate rotated surfaces.

In this paper, we have studied $m$-quasi-Einstein spacetimes. Some basic results of such spacetimes are derived. Perfect and viscous fluid $m$-quasi-Einstein spacetimes are also studied and the expressions of pressure, cosmological constant and energy density are obtained. We have proved that if the generator $\rho$ of an $m$-quasi-Einstein spacetime is a Killing vector field, then the spacetime is either conformally flat or of Petrov-type $N$. It is also shown that if the function $f$ of an $m$-quasi-Einstein spacetime satisfying Einstein’s equation is harmonic and the matter distribution is perfect fluid, then Segre’ characteristics of the Ricci tensor is [(1,1), 1]. Finally, an example is constructed for the proper existence of such a spacetime.

In this paper, using the Finslerian settings, we study the existence of parallel one forms (or, equivalently parallel vector fields) on a Riemannian manifold. We show that a parallel one form on a Riemannian manifold $M$ is a holonomy invariant function on the tangent bundle $TM$ with respect to the geodesic spray. We prove that if the metrizability freedom of the geodesic spray of $(M,F)$ is $1$, then the $(M,F)$ does not admit a parallel one form. We investigate a sufficient condition on a Riemannian manifold to admit a parallel one form. As by-product, we relate the existence of a proper affine Killing vector field by the metrizability freedom. We establish sufficient conditions for the existence of a parallel one form on a Finsler manifold. By counter-examples, we show that if the metrizability freedom is greater than 1, then the manifold (Riemannian or Finslerian) does not necessarily admit a parallel one form. Various special cases and examples are studied and discussed.

In this paper, we introduce the dual magnetic trajectories traced by the dual Frenet vectors of any dual curves in dual space via the dual Lorentz force, and then we give the characterizations of dual magnetic curves. Moreover, we define the dual flux ruled surfaces corresponding to the Killing vectors on the dual unit sphere and identify the conditions for these surfaces to be minimal or developable. Additionally, we present certain characterizations related to these surfaces. We demonstrate the visual representations of some dual flux ruled surfaces in some examples.

The evolution of some geometric quantities on a compact Riemannian manifold $M^n$ whose metric is Yamabe soliton is discussed. Using these quantities, lower bound on the soliton constant is obtained. We discuss about commutator of soliton vector fields. Also, the condition of soliton vector field to be a geodesic vector field is obtained.

In this paper we give a survey on ordinary helices which are integral curves of Killing vector fields on symmetric spaces of rank one. On a real space form ℝⁿ, Sⁿ or Hⁿ, all ordinary helices are generated by some Killing vector fields, and they are congruent each other if they have the same curvatures. But the situation is not the same for other symmetric spaces of rank one. Even a complex hyperbolic space admits bounded ordinary helices. We also make mention of an example of closed ordinary helices in a complex projective plane with 6 self-intersection points.