## 3-dimensional Lorentz space-forms and Seifert fiber spaces.(English)Zbl 0563.57004

A Lorentz space-form is a complete pseudo-Riemannian manifold of dimension 3 with a complete Lorentz metric of signature $$+--$$ and of constant curvature $$+1$$. A Lorentz space-form M is determined up to isometry by N/$$\Gamma$$, where N is a simply connected Lorentz space-form and $$\Gamma$$ is a discrete group of isometries acting freely on N.
A convenient model for N is the universal covering of PSL(2,$${\mathbb{R}})$$, denoted by $$S_{\infty}$$, where the Lorentz metric is realized from -1/8 (the Killing form). The full group of Lorentz isometries is a 6- dimensional Lie group with 4 connected components. This acts transitively on $$S_{\infty}$$ with the isotropy of e non-compact. This makes the analysis and classification of space-forms much more difficult than the usual Riemannian case because discrete subgroups of the group of isometries of $$S_{\infty}$$ do not necessarily act properly.
The space-forms are determined by studying the finitely generated discrete subgroups of $$Isometries(S_{\infty})$$ and those that act properly. A principal result is the following: If M is compact (or finite volume), then M must be orientable and possess a Seifert structure. In particular, M will have a finite Lorentz covering which is homeomorphic to a circle bundle over a closed surface of genus $$g>1$$ (or a product of punctured closed surface with $$S^ 1).$$
A space-form is called homogeneous if the group of Lorentz isometries acts transitively on M. The compact homogeneous Lorentz space-forms are classified in the paper. These turn out to be the compact quotients of the universal covering of PSL(2,$${\mathbb{R}})$$ by its cocompact discrete subgroups. Most compact Lorentz space-forms are not homogeneous. A four dimensional subgroup of the Lorentz isometries of $$S_{\infty}$$, called J, acts properly on $$S_{\infty}$$ and those M for which $$\Gamma$$ can be conjugated into J are called standard. (Incidently, J can also be identified with the full group of Riemannian isometries of a 3- dimensional twisted hyperbolic geometry on $$S_{\infty}.)$$ The following is the main result concerning compact standard Lorentz space-forms. A compact standard Lorentz space-form admits a compatible Seifert structure over a hyperbolic base and with non-zero Euler number. In particular, the $$S^ 1$$ fibers of the Seifert structure correspond to closed time-like geodesics of the Lorentz structure. Conversely, every closed orientable Seifert manifold over a hyperbolic base and with non-zero Euler number admits a standard Lorentz structure. Furthermore, the volume of a compact standard Lorentz manifold is an invariant of the underlying Seifert manifold structure and is a rational multiple of $$4\pi^ 2$$. The constructions are done explicitly.
It is shown that most compact Lorentz space-forms are standard. W. Goldman, using our results, will show in a subsequent work that (i) the Euler number of a compact Lorentz space-form can never be 0, and (ii) if M is not standard then it must be homeomorphic to a homogeneous space- form. Thus the topological classification of compact space-forms is complete. The isometric classification of the standard space-forms is carried out by the authors and K. B. Lee in the paper ”Deformation spaces for Seifert manifolds.”
Actually, in the present paper no torsion-free restrictions are made on $$\Gamma$$ (which is equivalent to requiring that $$\Gamma$$ acts freely) but instead the results are stated for finitely generated $$\Gamma$$ which act properly discontinuously. The claims for the resulting Lorentz orbifolds and Seifert orbifolds are analogous to those just cited for the space- forms.

### MSC:

 57N10 Topology of general $$3$$-manifolds (MSC2010) 57S30 Discontinuous groups of transformations 53C30 Differential geometry of homogeneous manifolds
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