Publ. Math. Debrecen 67/3-4 (2005), 471–480 On a class of Einstein space-time manifolds By ADELA MIHAI (Bucharest) and RADU ROSCA (Paris) Abstract. We deal with a general space-time (M,g) with usual differen- tiability conditions and hyperbolic metric g of index 1, which carries 3 skew- symmetric Killing vector fields X , Y , Z having as generative the unit time-like vector field e of the hyperbolic metric g. It is shown that such a space-time (M,g) is an Einstein manifold of curvature -1, which is foliated by space-like hypersur- faces M s normal to e and the immersion x : M s → M is pseudo-umbilical. In addition, it is proved that the vector fields X , Y , Z and e are exterior concur- rent vector fields and X , Y , Z define a commutative Killing triple, M admits a Lorentzian transformation which is in an orthocronous Lorentz group and the distinguished spatial 3-form of M is a relatively integral invariant of the vector fields X , Y and Z . 0. Introduction Let (M,g) be a general space-time with usual differentiability condi- tions and hyperbolic metric g of index 1. We assume in this paper that (M,g) carries 3 skew-symmetric Killing vector fields (abbr. SSK) X, Y , Z having as generative the unit time-like vector field e of the hyperbolic metric g (see [R1], [MRV]. Therefore, if ∇ is the Levi–Civita connection and ∧ means the wedge product of vector Mathematics Subject Classification: 53C15, 53D15, 53C25. Key words and phrases: space-time, skew-symmetric Killing vector field, exterior con- current vector field, orthocronous Lorentz group. This paper was written while the first author has visited Yamagata University, Faculty of Education, supported by a JSPS postdoctoral research fellowship. She would like to express her hearty thanks for the hospitality she received during this visit.