Twistor Bundle of a Neutral Kähler Surface

In this paper, we characterize neutral Kähler surfaces in terms of their positive twistor bundle. We prove that an O+,+(2,2)-oriented four-dimensional neutral semi-Riemannian manifold (M,g) admits a complex structure J with ΩJ∈⋀−M, such that (M,g,J) is a neutral-Kähler manifold if and only if the twistor bundle (Z1(M),gc) admits a vertical Killing vector field.

Conformal vector fields on almost co-Kähler manifolds

In this paper, we characterize almost co-Kähler manifolds with a conformal vector field. It is proven that if an almost co-Kähler manifold has a conformal vector field that is collinear with the Reeb vector field, then the manifold is a K-almost co-Kähler manifold. It is also shown that if a (κ, μ)-almost co-Kähler manifold admits a Killing vector field V, then either the manifold is K-almost co-Kähler or the vector field V is an infinitesimal strict contact transformation, provided that the (1,1) tensor h remains invariant under the Killing vector field.

A Note on Solitons with Generalized Geodesic Vector Field

We consider a general notion of an almost Ricci soliton and establish some curvature properties for the case in which the potential vector field of the soliton is a generalized geodesic or a 2-Killing vector field. In this vein, we characterize trivial generalized Ricci solitons.

Connecting and Closed Geodesics of a Kropina Metric

We prove some results about existence of connecting and closed geodesics in a manifold endowed with a Kropina metric. These have applications to both null geodesics of spacetimes endowed with a null Killing vector field and Zermelo’s navigation problem with critical wind.

A Note on Killing Calculus on Riemannian Manifolds

In this article, it has been observed that a unit Killing vector field ξ on an n-dimensional Riemannian manifold (M,g), influences its algebra of smooth functions C∞(M). For instance, if h is an eigenfunction of the Laplace operator Δ with eigenvalue λ, then ξ(h) is also eigenfunction with same eigenvalue. Additionally, it has been observed that the Hessian Hh(ξ,ξ) of a smooth function h∈C∞(M) defines a self adjoint operator ⊡ξ and has properties similar to most of properties of the Laplace operator on a compact Riemannian manifold (M,g). We study several properties of functions associated to the unit Killing vector field ξ. Finally, we find characterizations of the odd dimensional sphere using properties of the operator ⊡ξ and the nontrivial solution of Fischer–Marsden differential equation, respectively.

On m-quasi-Einstein spacetimes

In this paper, we have studied [Formula: see text]-quasi-Einstein spacetimes. Some basic results of such spacetimes are derived. Perfect and viscous fluid [Formula: see text]-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 [Formula: see text] of an [Formula: see text]-quasi-Einstein spacetime is a Killing vector field, then the spacetime is either conformally flat or of Petrov-type [Formula: see text]. It is also shown that if the function [Formula: see text] of an [Formula: see text]-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.

On Killing Vector Fields on Riemannian Manifolds

We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field w on a connected Riemannian manifold (M,g) we show that for each non-constant smooth function f∈C∞(M) there exists a non-zero vector field wf associated with f. In particular, we show that for an eigenfunction f of the Laplace operator on an n-dimensional compact Riemannian manifold (M,g) with an appropriate lower bound on the integral of the Ricci curvature S(wf,wf) gives a characterization of the odd-dimensional unit sphere S2m+1. Also, we show on an n-dimensional compact Riemannian manifold (M,g) that if there exists a positive constant c and non-constant smooth function f that is eigenfunction of the Laplace operator with eigenvalue nc and the unit Killing vector field w satisfying ∇w2≤(n−1)c and Ricci curvature in the direction of the vector field ∇f−w is bounded below by n−1c is necessary and sufficient for (M,g) to be isometric to the sphere S2m+1(c). Finally, we show that the presence of a unit Killing vector field w on an n-dimensional Riemannian manifold (M,g) with sectional curvatures of plane sections containing w equal to 1 forces dimension n to be odd and that the Riemannian manifold (M,g) becomes a K-contact manifold. We also show that if in addition (M,g) is complete and the Ricci operator satisfies Codazzi-type equation, then (M,g) is an Einstein Sasakian manifold.