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 \(G_i\).

设M是一个紧连通光滑伪黎曼流形，它通过等距允许拓扑传递G 作用，其中\(G = G_1 \ldots G_l\)是一个连通的半单李群，没有紧因数，其李代数为\({\ mathfrak {g}}= {\mathfrak {g}}_1 \oplus {\mathfrak {g}}_2 \oplus \cdots \oplus {\mathfrak {g}}_l\)。如果\（M_0，N_0，N_0 ^ I \）是的尺寸最大lightlike子空间相切中号，G ^，\（的G_i \） ，分别，然后我们研究ģ满足条件-actions \（M_0 = N_0 ^ 1 + \cdots + n_0^{l}\). 这个条件意味着轨道对于M上的伪黎曼度量是非退化的，这允许我们考虑轨道的法向丛。使用法向丛的特性到G轨道，我们通过考虑每个\(G_i\) 的自然度量来获得M的等距分裂。