Proceedings of the Japan Academy, Series A, Mathematical Sciences

Semisimple symmetric spaces without compact manifolds locally modelled thereon

Yosuke Morita

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Abstract

Let $G$ be a real reductive Lie group and $H$ a closed subgroup of $G$ which is reductive in $G$. In our earlier work it was shown that, if the homomorphism $i : H^{\bullet}(\mathfrak{g}_{\mathbf{C}}, \mathfrak{h}_{\mathbf{C}}; \mathbf{C}) \to H^{\bullet}(\mathfrak{g}_{\mathbf{C}},(\mathfrak{k}_{H})_{\mathbf{C}}; \mathbf{C})$ is not injective, there does not exist a compact manifold locally modelled on $G/H$. In this paper, we give a classification of the semisimple symmetric spaces $G/H$ for which $i$ is not injective. We also study the case when $G$ cannot be realised as a linear group.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 91, Number 2 (2015), 29-33.

Dates
First available in Project Euclid: 2 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.pja/1422885266

Digital Object Identifier
doi:10.3792/pjaa.91.29

Mathematical Reviews number (MathSciNet)
MR3310968

Zentralblatt MATH identifier
1320.53059

Subjects
Primary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 57S30: Discontinuous groups of transformations
Secondary: 17B56: Cohomology of Lie (super)algebras 22F30: Homogeneous spaces {For general actions on manifolds or preserving geometrical structures, see 57M60, 57Sxx; for discrete subgroups of Lie groups, see especially 22E40} 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]

Keywords
Local model $(G,X)$-structure Clifford–Klein form symmetric space relative Lie algebra cohomology invariant polynomial

Citation

Morita, Yosuke. Semisimple symmetric spaces without compact manifolds locally modelled thereon. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 2, 29--33. doi:10.3792/pjaa.91.29. https://projecteuclid.org/euclid.pja/1422885266


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