Deformation of compact Clifford-Klein forms of indefinite-Riemannian homogeneous manifolds
A Clifford-Klein form of a homogeneous manifold \(G/H\) is the double coset space \(\Gamma \setminus G/H\), where \(\Gamma\) is a subgroup of \(G\) acting properly discontinuously and freely on \(G/H\). Under the assumption that both \(G\) and \(H\) are non-compact real reductive Lie groups, we study the deformation of a compact Clifford-Klein form \(\Gamma \setminus G/H\) constructed in [\textit{T. Kobayashi}, Math. Ann. 285, 249-263 (1989; Zbl 0672.22011)], which is naturally an indefinite-Riemannian manifold. Surprisingly, in contrast to the Selberg-Weil local rigidity theorem in the Riemannian case with \(H\) compact, local rigidity can fail even in higher dimensional cases if \(H\) is non-compact, as was observed in [\textit{T. Kobayashi}, J. Geom. Phys. 12, 133-144 (1993; Zbl 0815.57029)]. This paper studies the local structure of the deformation of a compact Clifford-Klein form together with its quantitative estimate of the deformation parameter. In particular, we prove affirmatively a conjecture of \textit{W. Goldman} [J. Diff. Geom. 21, 301-308 (1985; Zbl 0591.53051)] in a general setting and give new compact Clifford-Klein forms of indefinite-Riemannian homogeneous manifolds. The proof is based on the criterion of properly discontinuous actions obtained independently by \textit{Y. Benoist} [Ann. Math. 144, 315-347 (1996; Zbl 0868.22013)] and \textit{T. Kobayashi} [J. Lie Theory 6, 147-163 (1996; Zbl 0863.22010)].
scientific article; zbMATH DE number 1143006