Compact Clifford-Klein forms of symmetric spaces -- revisited
Compact Clifford-Klein forms of symmetric spaces -- revisited
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DOI：
10.4310/pamq.2005.v1.n3.a6
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发表时间：
2005-09
期刊：
影响因子：
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通讯作者：
Toshiyuki Kobayashi;T. Yoshino
中科院分区：
文献类型：
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作者：
Toshiyuki Kobayashi;T. Yoshino
本文讨论了由真不连续群构成的对称空间的紧商的存在性问题，重点讨论了非黎曼情形。如果$H$是非紧的，则不连续群在齐次空间$G/H$中并不总是丰富的。文章的前半部分阐明了研究$G/H$的不连续群的一般机制，然后是具有/不具有紧商的对称空间的最新和最完整的列表。在后半部分，作为一般理论的应用，我们证明了：(I)存在具有常曲率的签名$(7，8)$的15维紧致伪黎曼流形；(Ii)存在1，3，7维复球面的紧商；(Iii)存在签名$(p，q)$的切空间形式的紧商当且仅当$p$小于$q的Hurwitz-Radon数。
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 $G/H$ if $H$ is non-compact. The first half of the article elucidates general machinery to study discontinuous groups for $G/H$, 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 if $p$ is smaller than the Hurwitz-Radon number of $q$.