Curvature properties of Gödel metric

Abstract

We present some corrections of a part of the original paper [5, Sec. 4, lines 14 9-15 9 ]. In addition, we give examples of manifolds related to the presented corrections. Further, we denote by S and κ the Ricci tensor and the scalar curvature of the product manifold (M × N , g = g × g), respectively. It is obvious that rank S = rank S and κ = κ on U C ⊂ M × N. Now [5, Eq. (24)] yields on this set rank S − κ n − 1 − (n − 2)L C g ≤ 2.

On geodesic mappings in particular class of Roter spaces

“…The tensor Q(A, T ) is called the Tachibana tensor of the tensors A and T , in short the Tachibana tensor (see, e.g., [17,21,23,26,27,35]). Thus, among other things, we have the (0, 6)-tensors:…”

“…Thus in particular, the Schwarzschild spacetime, the Kottler spacetime and the Reissner-Nordström spacetime satisfy (2.7). Recently, manifolds satisfying (2.7) were investigated among others in [17,23,35]. Warped product manifolds M × F N, of dimension 4, satisfying on…”

“…The warped product manifold M × F N , with 2-dimensional base (M , g) and (n − 2)-dimensional space of constant curvature ( N , g), n 4, is a manifold satisfying (2.7) and (2.8) [23,Theorem 7.1 (i)]. We refer to [8,15,17,18,21,23,27,35,38,61,66] for details on semi-Riemannian manifolds satisfying (2.5) and (2.6)-(2.8), as well as other conditions of this kind, named pseudosymmetry type curvature conditions or pseudosymmetry type conditions. It seems that (2.5) is the most important condition of that family of curvature conditions (see, e.g., [23]).…”

“…Quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations and the investigation on quasiumbilical hypersurfaces of conformally flat spaces (see, e.g., [18,23] and references therein). Quasi-Einstein manifolds satisfying some pseudosymmetry type conditions were investigated among others in [1,8,17,21,35]. Quasi-Einstein hypersurfaces in semi-Riemannian spaces of constant curvature were studied among others in [19,32,36,45], see also [4,Chapter 6.2], [5,Chapter 19.5], [6, Chapter 4.6], [18,68] and references therein.…”

Hypersurfaces in space forms satisfying some generalized Einstein metric condition

“…Thus in particular, the Schwarzschild spacetime, the Kottler spacetime and the Reissner-Nordström spacetime satisfy (1.9). Recently, manifolds satisfying (1.9) were investigated among others in [27,34,43].…”