We prove that a four-dimensional Lorentzian manifold that is curvature homogeneous of order 3, or CH3 for short, is necessarily locally homogeneous. We also exhibit and classify four-dimensional Lorentzian, CH2 manifolds that are not homogeneous. The resulting metrics belong to the class of null electromagnetic radiation, type N solutions on an anti-de Sitter background. These findings prove th...

In this paper we prove some results on the number of geodesics connecting two points or two submanifolds on a non-reversible complete Finsler manifold, in particular for complete Randers metrics. We apply the abstract results to the study of light rays and timelike geodesics with fixed energy on a standard stationary Lorentzian manifold.

1.1 Preliminaries. Let (L, gL μν) be a Lorentzian manifold which is supposed to be globally hyperbolic. This manifold is said (locally) static whenever it admits a (local) time-like Killing vector ∂t normal to a Cauchy surface Σ. In other words, the manifold admits a (local) coordinate frame (x0, x1, x2, x3) ≡ (t, ~x), where g0i = 0 (i = 1, 2, 3) and ∂tgμν = 0. Let us consider a real scalar fie...

Helfer in [6] was the first to produce an example of a spacelike Lorentzian geodesic with a continuum of conjugate points. In this paper we show the following result: given an interval [a, b] of IR and any closed subset F of IR contained in ]a, b], then there exists a Lorentzian manifold (M, g) and a spacelike geodesic γ : [a, b] → M such that γ(t) is conjugate to γ(a) along γ iff t ∈ F .

1.1 Preliminaries. Let (L, gL μν) be a Lorentzian manifold which is supposed to be globally hyperbolic. This manifold is said (locally) static whenever it admits a (local) time-like Killing vector ∂t normal to a Cauchy surface Σ. In other words, the manifold admits a (local) coordinate frame (x0, x1, x2, x3) ≡ (t, ~x), where g0i = 0 (i = 1, 2, 3) and ∂tgμν = 0. Let us consider a real scalar fie...

New splitting theorems in a semi-Riemannian manifold which admits an irrotational vector field (not necessarily a gradient) with some suitable properties are obtained. According to the extras hypothesis assumed on the vector field, we can get twisted, warped or direct decompositions. Some applications to Lorentzian manifold are shown and also S ×L type decomposition is treated.

We show in this note that if a twistor spinor has a zero on a Lorentzian spin manifold M of arbitrary dimension then the twistor is almost everywhere on M locally conformally equivalent to a parallel spinor.

In this paper, we study certain curvature conditions satisfying by the conharmonic curvature tensor in a Lorentzian para-Sasakian manifold with respect to the quarter-symmetric metric connection. AMS Mathematics Subject Classification (2010): 53B05, 53C25