A geometric splitting theorem for actions of semisimple Lie groups

Let M be a compact connected smooth pseudo-Riemannian manifold that admits a topologically transitive G-action by isometrics, where G = G(1) ... G(l) is a connected semisimple Lie group without compact factors whose Lie algebra is g = g(1) circle plus g(2) circle plus ... circle plus g(l). If m(0), n(0), n(0)(i) are the dimensions of the maximal lightlike subspaces tangent to M, G, G(i), respectively, then we study G-actions that satisfy the condition m(0) = n(0)(1) + ... + n(0)(l). This condition implies that the orbits are non-degenerate for the pseudo Riemannian metric on M and this allows us to consider the normal bundle to the orbits. Using the properties of the normal bundle to the G-orbits we obtain an isometric splitting of M by considering natural metrics on each G(i).