Kobayashi, Toshiyuki; Yoshino, Taro Compact Clifford-Klein forms of symmetric spaces – revisited. (English) Zbl 1145.22011 Pure Appl. Math. Q. 1, No. 3, 591-663 (2005). The existence problem of a compact quotient of a symmetric space by a properly discontinuous group with emphasis on the non-Riemannian case is studied. So, recent progress concerning the problem of the existence of compact Clifford-Klein forms in a symmetric space is discussed. In the first half of the paper the general machinery to study discontinuous groups for a homogeneous space is discussed. A most update and complete list of symmetric spaces with/without compact quotients is also presented. As applications of the general theory, in the second half the authors prove the existence of: (i) a 15-dimensional compact pseudo-Riemannian manifold of signature (7,8) with constant curvature; (ii) a compact quotient of the complex sphere of dimensions 1, 3 and 7; and (iii) a compact quotient of the tangential space form of signature \((p,q)\) if and only if \(p\) is smaller than the Hurwitz-Radon number of \(q\). Reviewer: Gheorghe Pitiş (Braşov) Cited in 1 ReviewCited in 38 Documents MSC: 22F30 Homogeneous spaces 22E40 Discrete subgroups of Lie groups 53C30 Differential geometry of homogeneous manifolds 53C35 Differential geometry of symmetric spaces 57S30 Discontinuous groups of transformations Keywords:discontinuous group; Clifford-Klein form; symmetric space; space form; pseudo-Riemannian manifold; discrete subgroup; uniform lattice; indefinite Clifford algebra PDFBibTeX XMLCite \textit{T. Kobayashi} and \textit{T. Yoshino}, Pure Appl. Math. Q. 1, No. 3, 591--663 (2005; Zbl 1145.22011) Full Text: DOI arXiv