## Deformation of compact Clifford-Klein forms of indefinite-Riemannian homogeneous manifolds.(English)Zbl 0891.22014

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 [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 [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 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 Y. Benoist [Ann. Math. 144, 315-347 (1996; Zbl 0868.22013)] and T. Kobayashi [J. Lie Theory 6, 147-163 (1996; Zbl 0863.22010)].

### MSC:

 22E46 Semisimple Lie groups and their representations 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics 57S30 Discontinuous groups of transformations 58H15 Deformations of general structures on manifolds 32G07 Deformations of special (e.g., CR) structures
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