The signature in actions of semisimple Lie groups on pseudo-Riemannian manifolds (Q2889741)

Let \(G\) denote a semisimple Lie group. It is known that there is a bijective correspondence between Ad(G)-invariant nondegenerate symmetric bilinear forms on the corresponding Lie algebra \(\mathfrak g\) and bi-invariant pseudo-Riemannian metrics on \(G\). Under such correspondence a bilinear form on \(\mathfrak g\) which is not a multiple of the Killing-Cartan form defines a pseudo-Riemannian metric on \(G\) that might be expected to provide a geometry that differs from that given by the Killing-Cartan form. The author shows that every bi-invariant pseudo-Riemannian metric on \(G\) is a finite sum of Killing-Cartan forms. The second goal is to obtain an estimate between the signature of \(G\) and that of a connected compact manifold \(M\) on which \(G\) acts.