On \(n\)-dimensional Lorentz manifolds admitting an isometry group of dimension \(n(n-1)/2+1\), for n\(\geq 4\) (Q1085822)

We have the following result of Obata: ''Let G be a connected Lie group of dimension r and H a compact subgroup of dimension (r-n). Assume that \(n(n-1)/2<\tau <n(n+1)/2\), \(n\geq 3\), \(n\neq 4\) and G acts almost effectively on G/H as a transformation group. Then G is of dimension \(n(n-1)/2+1\) and G/H is one of the spaces \[ C^ 1_ 0\times C_+^{n-1},\quad C^ 1_ 0\times C_-^{n-1},\quad C^ n_ 0,\quad C_-^ n, \] as a Riemannian manifold. Here we denote by \(C_+^ m,C_-^ m\) and \(C_ 0^ m\) an m-dimensional manifold of constant curvature. The author considers Lorentz manifolds and obtains a similar result: Theorem. Let M be a connected n-dimensional Lorentz manifold admitting a connected isometry group G of dimension \(n(n-1)/2+1\) (n\(\geq 4)\), whose isotropy subgroup is compact. Then M must be one of the spaces: \[ R\times N,\quad S^ 1\times N,\quad R\times P^{n-1},\quad S^ 1\times P^{n-1},\quad U^+_ n \] where N is a simply connected (n-1)- Riemannian manifold of constant curvature, \(P^{n-1}\) denotes real projective space, and \(U^+_ n=\{(u_ 1,...,u_ n)\), \(u_ n>0\}\) has a metric of the form \(ds^ 2=du^ 2_ 1+...+du^ 2_{n-1}-du^ 2_ n/(cu_ n)^ 2\), \(c>0\).