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 isometries, where \(G = G_1 \ldots G_l\) is a connected semisimple Lie group without compact factors whose Lie algebra is \({\mathfrak {g}}= {\mathfrak {g}}_1 \oplus {\mathfrak {g}}_2 \oplus \cdots \oplus {\mathfrak {g}}_l\). 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 \(m_0=n_0^1 + \cdots + n_0^{l}\). 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 \(G_i\).