For a homogeneous space G/H of reductive type, we consider the tangential homogeneous space G𝜃/H𝜃. In this paper, we give obstructions to the existence of compact Clifford–Klein forms for such tangential symmetric spaces and obtain new tangential symmetric spaces which do not admit compact Clifford–Klein forms. As a result, in the class of irreducible classical semisimple symmetric spaces, we have only two types of symmetric spaces which are not proved not to admit compact Clifford–Klein forms. The existence problem of compact Clifford–Klein forms for homogeneous spaces of reductive type, which was initiated by Kobayashi in 1980s, has been studied by various methods but is not completely solved yet. On the other hand, the one for tangential homogeneous spaces has been studied since 2000s and an analogous criterion was proved by Kobayashi and Yoshino. In concrete examples, further works are needed to verify Kobayashi–Yoshino’s condition by direct calculations. In this paper, some easy-to-check necessary conditions ( =obstructions) for the existence of compact quotients in the tangential setting are given, and they are applied to the case of symmetric spaces. The conditions are related to various fields of mathematics such as associated pair of symmetric space, Calabi–Markus phenomenon, trivializability of vector bundle (parallelizability, Pontrjagin class), Hurwitz–Radon number and Pfister’s theorem (the existence problem of common zero points of polynomials of odd degree).

中文翻译：

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切向对称空间紧致 Clifford-Klein 形式存在的障碍

对于同质空间G/H归约型，我们考虑切向齐次空间G𝜃/H𝜃. 在本文中，我们为这种切向对称空间的紧致 Clifford-Klein 形式的存在提供了障碍，并获得了不允许紧致 Clifford-Klein 形式存在的新的切向对称空间。因此，在不可约的经典半单对称空间类中，我们只有两种类型的对称空间，它们没有被证明不承认紧致 Clifford-Klein 形式。小林在 1980 年代提出的归约型齐次空间的紧克利福德-克莱因形式的存在性问题，已经通过各种方法进行了研究，但尚未完全解决。另一方面，自 2000 年代以来一直在研究切向齐次空间，并且小林和吉野证明了类似的标准。在具体的例子中，需要进一步的工作来通过直接计算来验证 Kobayashi-Yoshino 的状态。在本文中，一些易于检查的必要条件（ =给出了在切线设置中存在紧商的障碍），并将它们应用于对称空间的情况。这些条件与数学的各个领域有关，例如关联的对称空间对、Calabi-Markus 现象、向量丛的可平凡性（并行性、Pontrjagin 类）、Hurwitz-Radon 数和 Pfister 定理（多项式的公共零点的存在性问题）奇数度）。