The isometry group of a compact Lorentz manifold. I (Q1365479)

The authors classify the connected Lie groups that may act isometrically on compact Lorentz manifolds. The main result is Theorem 11.1: Let \(G\) be a connected Lie group. Then the following are equivalent: (1) There exists a compact connected Lorentz manifold \(M\) on which \(G\) acts locally faithfully and isometrically; and (2) \(G\) is isomorphic to \(L\times K\times \mathbb{R}^d\), where \(K\) is compact and semisimple (or trivial), \(d \geq 0\), and \(L\) is in the following list: (a) \(\widetilde {SL} (2,\mathbb{R})\), (b) \(\text{Aff} (\mathbb{R})\), (c) a Heisenberg group \(H_n\), (d) a certain countable family of semidirect products \(\mathbb{R} \ltimes H_n\), (e) the trivial group \(\{e\}\). Moreover, if \(L\) is in the above list, then any locally faithful, isometric action of \(L\) on a compact Lorentz manifold is locally free. [Part II is reviewed below as Zbl 0897.53047].