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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Riemannian $ 4$-symmetric spaces


Author: J. A. Jiménez
Journal: Trans. Amer. Math. Soc. 306 (1988), 715-734
MSC: Primary 53C30; Secondary 53C35
DOI: https://doi.org/10.1090/S0002-9947-1988-0933314-6
MathSciNet review: 933314
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Abstract: The main purpose of this paper is to classify the compact simply connected Riemannian $ 4$-symmetric spaces. As homogeneous manifolds, these spaces are of the form $ G/L$ where $ G$ is a connected compact semisimple Lie group with an automorphism $ \sigma $ of order four whose fixed point set is (essentially) $ L$. Geometrically, they can be regarded as fiber bundles over Riemannian $ 2$-symmetric spaces with totally geodesic fibers isometric to a Riemannian $ 2$-symmetric space. A detailed description of these fibrations is also given. A compact simply connected Riemannian $ 4$-symmetric space decomposes as a product $ {M_1} \times \ldots \times {M_r}$ where each irreducible factor is: (i) a Riemannian $ 2$-symmetric space, (ii) a space of the form $ \{ U \times U \times U \times U\} /\Delta U$ with $ U$ a compact simply connected simple Lie group, $ \Delta U = $ diagonal inclusion of $ U$, (iii) $ \{ U \times U\} /\Delta {U^\theta }$ with $ U$ as in (ii) and $ {U^\theta }$ the fixed point set of an involution $ \theta $ of $ U$, and (iv) $ U/K$ with $ U$ as in (ii) and $ K$ the fixed point set of an automorphism of order four of $ U$. The core of the paper is the classification of the spaces in (iv). This is accomplished by first classifying the pairs $ (\mathfrak{g},\,\sigma )$ with $ \mathfrak{g}$ a compact simple Lie algebra and $ \sigma $ an automorphism of order four of $ \mathfrak{g}$. Tables are drawn listing all the possibilities for both the Lie algebras and the corresponding spaces. For $ U$ "classical," the automorphisms $ \sigma $ are explicitly constructed using their matrix representations. The idea of duality for $ 2$-symmetric spaces is extended to $ 4$-symmetric spaces and the duals are determined. Finally, those spaces that admit invariant almost complex structures are also determined: they are the spaces whose factors belong to the class (iv) with $ K$ the centralizer of a torus.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0933314-6
Article copyright: © Copyright 1988 American Mathematical Society

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