Riemannian -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 -symmetric spaces. As homogeneous manifolds, these spaces are of the form where is a connected compact semisimple Lie group with an automorphism of order four whose fixed point set is (essentially) . Geometrically, they can be regarded as fiber bundles over Riemannian -symmetric spaces with totally geodesic fibers isometric to a Riemannian -symmetric space. A detailed description of these fibrations is also given. A compact simply connected Riemannian -symmetric space decomposes as a product where each irreducible factor is: (i) a Riemannian -symmetric space, (ii) a space of the form with a compact simply connected simple Lie group, diagonal inclusion of , (iii) with as in (ii) and the fixed point set of an involution of , and (iv) with as in (ii) and the fixed point set of an automorphism of order four of . The core of the paper is the classification of the spaces in (iv). This is accomplished by first classifying the pairs with a compact simple Lie algebra and an automorphism of order four of . Tables are drawn listing all the possibilities for both the Lie algebras and the corresponding spaces. For "classical," the automorphisms are explicitly constructed using their matrix representations. The idea of duality for -symmetric spaces is extended to -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 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