Nonstandard Lorentz space forms. (English) Zbl 0591.53051

Goldman, William M.

J. Differ. Geom. 21, 301-308 (1985).

This paper is a sequel to the paper ”Three-dimensional Lorentz space forms and Seifert fiber spaces” by F. Raymond and R. Kulkarni [ibid., 231-268 (1985; Zbl 0563.57004)] and answers two questions raised in a previous version of that paper. The subject of both papers are Lorentz metrics on 3-dimensional manifolds having constant nonzero curvature. Such a structure will henceforth be called a ”Lorentz structure” for brevity, and corresponds to a Lorentz metric locally isometric to SL(2,\({\mathbb{R}})\) and its bi-invariant Lorentz metric defined by the Killing form. Kulkarni and Raymond construct a family of Lorentz structures on a certain class of Seifert fiber spaces which satisfy a strong additional property that they call ”standard”: a Lorentz structure is standard if it possesses a timelike Killing vector field. (It is not difficult to prove that a standard Lorentz structure on a closed manifold must be complete.) The first result of the present paper demonstrates the existence of complete Lorentz structures on certain closed 3-manifolds which are not standard. It follows from the construction of these examples that the deformation space of complete Lorentz structures on such closed 3-manifolds is not Hausdorff.
A major result of the Kulkarni-Raymond paper is that a closed 3-manifold which admits a complete Lorentz structure must be Seifert-fibered over a hyperbolic base. The second result of the present paper is that the Euler class of the Seifert fibering is nonzero, i.e. the 3-manifold is not covered by a product of a surface with a circle. Thus a closed 3-manifold which admits a complete Lorentz structure must admit a standard Lorentz structure. It follows that the class of closed 3-manifolds which admit complete Lorentz structures is exactly the class of Seifert fiber spaces whose base is a hyperbolic orbifold and such that the Euler class of the Seifert fibering is nonzero. This class of 3-manifolds constitutes one of Thurston’s eight classes of geometric 3-manifolds.

MSC:

53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
57R15	Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57N10	Topology of general \(3\)-manifolds (MSC2010)

Keywords:

Killing vector field; deformation of geometric; structure; Seifert fiber spaces; Lorentz structure; closed 3-manifold; Euler class

Citations:

Zbl 0575.57002; Zbl 0563.57004