2021/06/06 by Koichi Tojo, Tojo, Koichi
Biochemistry, Genetics and Molecular Biology · Mathematics · #53C30 (Secondary) #53C35 #57S30 (Primary) #Advanced Algebra and Geometry #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #Microtubule and mitosis dynamics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2106.03250
openalex publication_date 2021/06/06 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
For a homogeneous space G/H of reductive type, we consider the tangential\nhomogeneous space G_\θ/H_\θ. In this paper, we give obstructions to\nthe existence of compact Clifford-Klein forms for such tangential symmetric\nspaces and obtain new tangential symmetric spaces which do not admit compact\nClifford-Klein forms. As a result, in the class of irreducible classical\nsemisimple symmetric spaces, we have only two types of symmetric spaces which\nare not proved not to admit compact Clifford-Klein forms.\n The existence problem of compact Clifford-Klein forms for homogeneous spaces\nof reductive type, which was initiated by T. Kobayashi in 1980s, has been\nstudied by various methods but is not completely solved yet. On the other hand,\nthe one for tangential homogeneous spaces has been studied since 2000s and an\nanalogous criterion was proved by T. Kobayashi and T. Yoshino. In concrete\nexamples, further works are needed to verify Kobayashi-Yoshino's condition by\ndirect calculations. In this paper, some easy-to-check necessary\nconditions(=obstructions) for the existence of compact quotients in the\ntangential setting are given, and they are applied to the case of symmetric\nspaces. The conditions are related to various fields of mathematics such as\nassociated pair of symmetric space, Calabi-Markus phenomenon, trivializability\nof vector bundle (parallelizability, Pontrjagin class), Hurwitz-Radon number\nand Pfister's theorem (the existence problem of common zero points of\npolynomials of odd degree).\n