Locally free actions on Lorentz manifolds (Q1584715)

The paper studies special types of actions of Lie groups \(G\) on Lorentz manifolds \(M\). The main theorem reads as follows. Let \(G\) be a connected Lie group which locally faithfully and orbit nonproperly acts by Lorentz isometries on a connected Lorentz manifold \(M\); assume that every connected stabilizer is compact. Then 1. the center of \(G\) is noncompact, or 2. \(\text{Ad }{\mathfrak g}(G)\) is not closed in \(\text{GL}({\mathfrak g})\) or 3. the Lie algebra \({\mathfrak g}\) to \(G\) has a one-dimensional ideal, or 4. \(\text{sl}(2,\mathbb{R})\) is a direct summand of \({\mathfrak g}\). Here an action is called orbit nonproper iff for some \(m_0\in M\) the map \(G\to M\), \(g\mapsto gm_0\) is non-proper. The paper also presents kinds of logical conversions of the main theorem, some Riemannian analog, and more variations on the theme. Some results which appear as preparatory or miscellaneous are interesting by themselves.