Nonstandard Lorentz space forms

Abstract

In their recent paper [8], Kulharni and Raymond show that a closed 3-manifold which admits a complete Lorentz metric of constant curvature 1 (henceforth called a complete Lorentz structure) must be Seifert fibered over a hyperbolic base. Furthermore on every such Seifert fibered 3-manifold with nonzero Euler class they construct such a Lorentz metric. Moreover the Lorentz structure they construct has a rather strong additional property, which they call standard: A Lorentz structure is standard if its causal double cover possesses a timelike Killing vector field. Equivalently, it possesses a Riemannian metric locally isometric to a left-invariant metric on SL(2, R). Kulkarni and Raymond asked if every closed 3-dimensional Lorentz structure is standard. This paper provides a negative answer to this question (Theorem 1) and a positive answer to the implicit question raised in [8, 7.1.1] (Theorem 3). Theorem 1. Let M3 be a closed 3-manifold which admits a homogeneous Lorentz structure and satisfies Hι{M R) Φ 0. Then there exists a nonstandard complete Lorentz structure on M. In [8] it is shown that the unit tangent bundle of a closed surface admits a homogeneous Lorentz structure. Therefore we obtain: Corollary 2. There exists a complete Lorentz structure on the unit tangent bundle of any closed surface F of genus greater than one which is not standard. The homogeneous Lorentz structures are all classified in [8]. A circle bundle of Euler number j over a closed surface F, χ(F) < 0, has a homogeneous structure if and only if j\χ(F) (an analogous statement holds when M has singular fibers, i.e. when F is an orbifold). We also show: Theorem 3. Let M3 be a 3-manifold which admits a complete Lorentz structure. Then M3 is not covered by a product F X Sι, Fa closed surface. Theorem 3 implies that the Euler class of the Seifert fiber structure of M3 is nonzero.