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**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.

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 |