Homogeneous Lorentz manifolds with isotropy subgroup \(U(2)\) or \(SO(2)\) (Q1202806)

In previous papers the author gave several results to classify connected \(n\)-dimensional Lorentz manifolds admitting a connected isometry group of certain specified dimensions and whose isotropy subgroup is compact [Hokkaido Math. J. 15, 309-315 (1986; Zbl 0608.53057), Proc. Am. Math. Soc. 100, 329-334 (1987; Zbl 0621.53049), Tsukuba J. Math. 13, No. 1, 113-129 (1989; Zbl 0679.53053)]. For example \(n\)-dimensional \((n\geq 5)\) Lorentz manifolds admitting an isometry group of dimension \(n(n-1)/2+1\) are classified in the second paper listed above. In this paper the classifications of certain cases that were excluded in the above papers are stated and proved. The cases in the paper under review all have isotropy subgroup \(U(2)\) or \(SO(2)\). The three main results are as follows. Theorem A classifies all simply connected 5-dimensional Lorentz manifolds admitting a connected 9-dimensional isometry group with compact isotropy subgroup at every point. Theorem B classifies all simply connected 3-dimensional Lorentz manifolds admitting a connected 4- dimensional isometry group with compact isotropy subgroup at every point. Theorem C classifies all simply connected 4-dimensional Lorentz manifolds admitting a connected 5-dimensional isometry group with compact isotropy subgroup at every point.