×

zbMATH — the first resource for mathematics

Anti-de Sitter 3-manifolds with non-trivial Killing field. (Variétés anti-de Sitter de dimension 3 possédant un champ de Killing non trivial.) (French. Abridged English version) Zbl 0896.53043
This note classifies all compact 3-dimensional Lorentz manifolds of constant negative curvature which possess a nontrivial Killing field. Such manifolds are locally isometric to anti-de Sitter space \(X\), a convenient model of which is the universal covering group \(\widetilde{PSL} (2, \mathbb{R})\), with isometries given by the group \(G = \widetilde{PSL} (2, \mathbb{R})\times\widetilde{PSL}(2, \mathbb{R})\) where one factor acts by left-multiplication and the other by right-multiplication. R. S. Kulkarni and F. Raymond [J. Differ. Geom. 21, 113-138 (1985; Zbl 0563.57004)] gave the first examples of compact anti-de Sitter manifolds. Their examples are quotients of anti-de Sitter space by discrete groups of the form \(\text{graph}(\rho)\), where \(\rho:\Gamma\longrightarrow\widetilde{PSL}(2, \mathbb{R})\) is a homomorphism taking values in a compact subgroup of \(\widetilde{PSL}(2, \mathbb{R})\), and \(\Gamma\subset\widetilde{PSL}(2, \mathbb{R})\) is a discrete group. (This condition is useful since it implies that the holonomy group lies in a subgroup \(H\subset G\) which acts transitively and properly on \(X\).) Furthermore, Kulkarni and Raymond showed that every properly discontinuous action with compact quotient is equivalent to the graph of a homomorphism \(\rho:\Gamma\longrightarrow\widetilde{PSL}(2,\mathbb{R})\) where \(\Gamma\subset\widetilde{PSL}(2,\mathbb{R})\) is a discrete cocompact subgroup. Subsequent further examples were given by W. M. Goldman [J. Differ. Geom. 21, 301-308 (1985; Zbl 0591.53051)] and E. Ghys [Ann. Sci. Ècole Norm. Sup. 20, 251-270 (1987; Zbl 0663.58025)], all of which possess nontrivial Killing vector fields, but the holonomy does not lie in a properly acting transitive subgroup of \(G\). The difficulty in these examples is that, in general, compactness of an indefinite metric does not imply completeness. Recently, B. Klingler [Math. Ann. 306, 353-370 (1996; Zbl 0862.53048)] proved that compact anti-de Sitter manifolds are geodesically complete; thus any compact Lorentzian manifold of constant negative curvature is a quotient of anti-de Sitter space. It follows that whenever \(\rho\) is sufficiently close to the trivial representation, \(\text{graph}(\rho)\) acts properly and cocompactly on \(X\).
The paper under review determines explicitly how small \(\rho\) must be for \(\text{graph}(\rho)\) to act properly, in the special case that \(\rho\) takes values in an abelian subgroup of \(\widetilde{PSL}(2, \mathbb{R})\). If \(\rho\) takes values in an elliptic or parabolic one-parameter subgroup, then \(\text{graph}(\rho)\) acts properly. Thus, suppose that \(\rho\) takes values in a hyperbolic one-parameter subgroup \(H\). For \(\gamma\in\Gamma\), let \(t(\gamma)\) denote the minimum displacement of \(\gamma\) acting on the hyperbolic plane. Then the stable norm on \(H^1(\Gamma,\mathbb R)\) is defined as the the function \[ \| h\| = \lim_{n\to\infty}{\textstyle \frac 1n} \inf \{t(\gamma_n) \mid [\gamma_n] = n h \}, \] where \([\gamma]\) denotes the equivalence class of \(\gamma\in\Gamma\) in \(H^1(\Gamma,\mathbb R)\). Identifying the \(H\)-valued representations \(\rho\) with \(H^1(\Gamma,\mathbb R)\), the author proves that \(\text{graph}(\rho)\) acts properly if and only if \(\|\rho\| < 1\). The author investigates sufficient conditions for the existence of oscillatory solutions of equations of the form \[ u''' + p_1(t)u'' + p_2(t)u'' + p_3(t)u = 0. \] This interesting survey includes explicitly recent results on oscillatory solutions. It is a continuation of an article of K.-H. Mayer [J. Oscillation 136, No. 5, 35-102 (1987; per bibl.)].

MSC:
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
PDF BibTeX XML Cite
Full Text: DOI