Completeness of certain pseudo-Riemannian (locally) symmetric spaces (Q5962659)

Let \(G\) be a Lie group and \((X, G)\) a homogenous \(G\)-space. A manifold \(M\) is called a \((G,X)\) manifold if it admits an atlas taking values in \(X\) such that all chart transition functions can be described by elements in \(G\). A \((G,X)\)-structure on \(M\) is called Kleinian if \(M\) is obtained as a quotient of some open subset \(U \subset X\) by the action of a free and properly discontinuous discrete subgroup \(\Gamma \subset G\).NEWLINENEWLINENEWLINEAs a consequence of the Hopf-Rinow theorem if \(X\) is Riemannian, all compact \((G,X)\)-manifolds are complete. It is an important observation that a similar result fails in the pseudo-Riemannian case; various examples and counterexamples are known and results covering special cases were established; some of them are described in the article under consideration.NEWLINENEWLINENEWLINEIn this important paper the author extends these results, establishing the following main theorems and describing some of their consequences:NEWLINENEWLINENEWLINETheorem 1. All compact Kleinian manifolds with a \((U(d-1,1)\ltimes \mathbb{C}^d, \mathbb{C}^d)\) are complete.NEWLINENEWLINENEWLINETheorem 3. Let \(L\) be a semi-simple Lie group of real rank \(1\) equipped with the natural action by \(L\times L\) given by \((g,h)\cdot l = glh^{-1}\). Then all compact Kleinian \((L\times L, L)\)-manifolds are complete.