Riemannian Space-times of Godel Type in Five Dimensions

The five-dimensional (5D) Riemannian Godel-type manifolds are examined in light of the equivalence problem techniques, as formulated by Cartan. The nec- essary and sufficient conditions for local homogeneity of these 5D manifolds are derived. The local equivalence of these homogeneous Riemannian manifolds is studied. It is found that they are characterized by two essential parameters m 2 and !: identical pairs (m 2 , !) correspond to locally equivalent 5D manifolds. An irreducible set of isometrically nonequivalent 5D locally homogeneous Riemannian Godel-type metrics are exhibited. A classification of these manifolds based on the essential parameters is presented, and the Killing vector fields as well as the cor- responding Lie algebra of each class are determined. It is shown that apart from the (m 2 = 4 ! 2 , ! 6 0) and (m 2 6 0, ! = 0) classes the homogeneous Rieman- nian Godel-type manifolds admit a seven-parameter maximal group of isometry (G7). The special class (m 2 = 4 ! 2 , ! 6 0) and the degenerated Godel-type class ( m 2 6 0, ! = 0) are shown to have a G9 as maximal group of motion. The break- down of causality in these classes of homogeneous Godel-type manifolds are also examined.