arXiv:1401.6920v1 [math.DG] 27 Jan 2014 CURVATURE PROPERTIES OF G ¨ ODEL METRIC RYSZARD DESZCZ, MARIAN HOTLO ´ S, JAN JE LOWICKI, HARADHAN KUNDU AND ABSOS ALI SHAIKH Abstract. The main aim of this article is to investigate the geometric structures admitting by the G¨ odel spacetime which produces a new class of semi-Riemannian manifolds (see Theorem 4.1 and Theorem 4.4). We also consider some extension of G¨ odel metric (see Example 4.1). 1. Introduction In 1949 G¨ odel [1] obtained an exact solution of Einstein field equation with a non-zero cosmological constant corresponding to a universe in rotation and with an incoherent matter distribution. In that paper he described a metric nowadays called G¨ odel metric as exact and stationary solution of Einstein field equation, which describes a rotating, homogeneous but non-isotropic spacetime. Possessing a series of strange properties, it remains still today quite interesting mathematically and significant physically. For example, it contains rotating matter but have not singularity, and also it is cyclic Ricci parallel [2]. It is known that the Weyl conformal tensor of the G¨ odel solution has Petrov type D, and G¨ odel solution is, up to local isometry, the only perfect fluid solution of Einstein field equation admitting five dimensional Lie algebra of Killing vectors. G¨ odel spacetime is geodesically complete, its timelike curves are closed [3]. Also G¨ odel spacetime is not globally hyperbolic but diffeomorphic to R 4 and is simply connected. G¨ odel metric is the Cartesian product of a factor R with a three dimensional Lorentzian manifold with signature (− + ++). G¨ odel metric and its properties have been studied by various authors to describe the G¨ odel universe. Kundt [4] studied its geodesics in 1956, and Hawking and Ellis [5] emphasized on coordinates showing its rotational symmetry to draw a nice picture of its dynamics in their book in 1973. Malament [6] calculated the minimal energy of a closed timelike curve of G¨ odel spacetime. In 2001 Radojevi´ c [7] presented modification of G¨ odel metric in order to find out some other perfect fluid solutions. Induced matter theory and embedding of G¨ odel universe 0 2010 Mathematics Subject Classification: 53B20, 53B30, 53B50, 53C50, 53C80, 83C57. Key words and phrases: G¨ odel spacetime, Weyl conformal curvature tensor, conharmonic curvature tensor, Tachibana tensor, pseudosymmetric manifold, pseudosymmetry type curvature condition, quasi-Einstein mani- fold. 1