On causal structure of homogeneous Lorentzian manifolds (Q911071)

Lorentzian manifolds with transitive actions of their groups of isometries are investigated. First it is shown that the 10 most common causality conditions reduce to 5 essential conditions in the case under consideration. More explicitly, `causally continuous' is shown to be equivalent to (past or) `future distinguishing'. Also two refined causality conditions are added: `homogeneous global hyperbolicity' and `uniform stable causality'. Then, the chronological future of the identity is described in terms of future time-like one-parameter subsemigroups of the underlying Lie group M. Conditions are given for M in terms of its Lie algebra in order to violate the chronology condition. Furthermore, sufficient conditions are developed for M to be homogeneous globally hyperbolic or uniform stably causal. Finally, in a longer section, the preceding investigations are applied to the description of the causal structure of all geodesically complete spaces given by left- invariant Lorentzian metrics on three-dimensional simply connected solvable Lie groups. For a discussion of sufficient conditions for M to be geodesically incomplete see the paper of \textit{D. V. Alekseevskij} and \textit{B. A. Putko} in Funct. Anal. Appl. 21, No.3, 73-74 (1987); translation from Funkts. Anal. Prilozh. 21, No.3, 73-74 (1987; Zbl 0631.53040)]. Unfortunately, though the paper under consideration presents a lot of interesting results, incorrect references and misunderstandable formulations make it difficult to read.