T. Kobayashi and T. Yoshino,

*Compact Clifford-Klein forms of
symmetric spaces — revisited*,

Pure and Appl. Math. Quarterly**1** (2005), 603-684,

Special Issue: In Memory of Armand Borel, DOI: 10.4310/PAMQ.2005.v1.n3.a6. math.DG/0509543..

Pure and Appl. Math. Quarterly

Special Issue: In Memory of Armand Borel, DOI: 10.4310/PAMQ.2005.v1.n3.a6. math.DG/0509543..

This article discusses the existence problem of a compact quotient of a symmetric space by a properly discontinuous group with emphasis on the non-Riemannian case.Discontinuous groups are not always abundant in a homogeneous space

G/HifHis non-compact. The first half of the article elucidates general machinery to study discontinuous groups forG/H, followed by the most update and complete list of symmetric spaces with/without compact quotients. In the second half, as applications of general theory, we prove: (i) there exists a 15 dimensional compact pseudo-Riemannian manifold of signature (7,8) with constant curvature, (ii) there exists a compact quotient of the complex sphere of dimension 1, 3 and 7, and (iii) there exists a compact quotient of the tangential space form of signature (p,q) if and only ifpis smaller than the Hurwitz-Radon number ofq.

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The original publication is available at International Press.

© Toshiyuki Kobayashi