Orbit nonproper dynamics on Lorentz manifolds
This carefully written and highly readable paper is the last of a long series by author and \textit{G. Stuck}, originally motivated by a paper of \textit{N. Kowalsky} [Ann. Math. (2), 144, No. 3, 611-640 (1996; Zbl 0871.53048)]. The author characterizes in it the connected Lie groups \(G\) with simply connected nilradical which admit an orbit nonproper action by isometries of some connected Lorentz manifold. Here, orbit nonproper means that for some point \(x\), the evaluation map \(g\mapsto g(x)\) is nonproper. Such an action exists if and only if \(G\) satisfies one of the following conditions: the adjoint action \(G\to \text{GL} ({\mathfrak g})\) is nonproper, or \({\mathfrak g}\) has a direct summand \({\mathfrak so} (n,i)\), \(i\leq 2\leq n\), or \({\mathfrak g}\) has a nonzero abelian ideal \({\mathfrak a}\) admitting a positive definite or Minkowski quadratic form such that the adjoint action of \({\mathfrak g}\) on \({\mathfrak a}\) is conformal with respect to this form. There is another, equivalent formulation of the third condition which can be checked directly by looking at the adjoint representation of a Levi factor on the center of the nilradical. The proof of the theorem depends on a detailed analysis of the structure of \({\mathfrak so}(n,1)\) and of the representations of \({\mathfrak sl}_2\) and of noncompact simple or reductive groups.
scientific article; zbMATH DE number 1734190