On a class of Einstein space-time manifolds

We deal with a general space-time (M,g) with usual differentiability 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 hypersurfaces 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 concurrent 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.