A characterization of Euclidean spheres out of connected, compact, Einstein Riemannian manifolds of constant scalar curvature is made by a characterization of a vector field with an eigenvalue equation for the Laplacian on vector fields.
This is a survey of the principal results about the geodesic completeness of nondegenerate hypersurfaces in Lorentzian manifolds from a structural point of view. Some of these results retain their validity in the case of semi-Riemannian submanifolds in semi-Euclidean spaces, as well.
Null isotropy in a spacetime is defined. The relation of null isotropy to the constant curvature and infinitesimal spatial isotropy is investigated. The influence of null isotropy on conjugate points along null geodesics and curvature singularities is investigated.Keywords Spacetime · Instantaneous observer · Stress-energy tensor · Perfect fluid · Einstein equation · Jacobi operator · Weyl operator · Null isotropic · Infinitesimally spatially isotropic
IntroductionA characterization of Robertson-Walker metrics was first made by Karcher [6] by introducing the concept of infinitesimal spatial isotropy. Later [2] and [7] obtained an equivalent characterization of those metrics by introducing the concept of infinitesimal null isotropy (which is equivalent to infinitesimal spatial isotropy). The concept of null isotropy (and then null anisotropy) was first introduced in [3] (in [4]) for semi-Riemannian manifolds (for spacetimes). The purpose of this paper is to study the influence of null isotropy on conjugate points along null geodesics and curvature singularities. Hence we first investigate the relation of null isotropy to the infinitesimal spatial isotropy in spacetimes. Notice that every infinitesimally spatially isotropic spacetime is null isotropic since it is conformally flat (cf. [6]). We first show that, a null vector u is isotropic iffW u = 0, where W is the Weyl tensor andW u is the Weyl operator onū ⊥ . Then, by making use of this observation, we show that, a null isotropic spacetime at q ∈ M is infinitesimally spatially isotropic for an instantaneous observer z ∈ T q M iffr u = λ ∈ R, ∀u ∈ χ + (z). Furthermore, z is unique iff λ = 0, and conversely, z is not unique iff λ = 0 F. Erkekoglu (B)
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