Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] P. J. Graham and A. J. Ledger, Sur un classe de $ s$-variétés Riemanniennes ou affines, C. R. Acad. Sci. Paris 267 (1968), 105-107. MR 0234380 (38:2697)
  • [2] -, $ s$-regular manifolds, Differential Geom. in honor of K. Yano, Kinoluniga, Tokyo, 1972, pp. 133-144. MR 0328825 (48:7167)
  • [3] A. Gray, Riemannian manifolds with geodesic symmetries of order 3, J. Differential Geom. 7 (1972), 343-369. MR 0331281 (48:9615)
  • [4] S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, London and New York, 1978. MR 514561 (80k:53081)
  • [5] R. Hermann, A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle, Proc. Amer. Math. Soc. 11 (1960), 236-242. MR 0112151 (22:3006)
  • [6] -, On the differential geometry of foliations, Ann. of Math. (2) 72 (1960), 445-457. MR 0142130 (25:5523)
  • [7] J. A. Jiménez, Riemannian $ 4$-symmetric spaces, Ph.D. Thesis, Univ. of Durham, Durham, England, 1983.
  • [8] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vols. I and II, Wiley-Interscience, New York, 1963 and 1969. MR 0152974 (27:2945)
  • [9] O. Kowalski, Riemannian manifolds with general symmetries, Math. Z. 136 (1974), 137-150. MR 0341346 (49:6097)
  • [10] -, Classification of generalized symmetric spaces of dimension $ < 5$, Rozpravy Československé Akad. Věd Řada Mat. Přírod. Věd. 85 (1975), 1-61.
  • [11] -, Existence of generalized symmetric Riemannian spaces of arbitrary order, J. Differential Geom. 12 (1977), 203-208. MR 0514787 (58:24128)
  • [12] -, Generalized symmetric spaces, Lecture Notes in Math., vol. 805, Springer-Verlag, Berlin and New York, 1980. MR 579184 (83d:53036)
  • [13] A. J. Ledger, Espace de Riemann symetriques généralisés, C. R. Acad. Sci. Paris 264 (1967), 947-948. MR 0221435 (36:4487)
  • [14] A. J. Ledger and M. Obata, Affine and Riemannian $ s$-manifolds, J. Differential Geom. 2 (1968), 451-459. MR 0244893 (39:6206)
  • [15] O. Loos, Spiegelungsräume und homogene symmetrische Räume, Math. Z. 99 (1967), 141-170. MR 0212742 (35:3608)
  • [16] -, Reflection spaces of minimal and maximal torsion, Math. Z. 106 (1968), 67-72. MR 0231322 (37:6877)
  • [17] J. A. Wolf, Locally symmetric homogeneous spaces, Comment. Math. Helv. 37 (1962), 65-101. MR 0148012 (26:5522)
  • [18] -, On the classification of Hermitian symmetric spaces, J. Math. Mech. 13 (1964), 489-496. MR 0160850 (28:4060)
  • [19] -, Spaces of constant curvature (5th ed.), Publish or Perish, Wilmington, Del., 1984. MR 928600 (88k:53002)
  • [20] J. A. Wolf and A. Gray, Homogeneous spaces defined by Lie group automorphisms. I, II, J. Differential Geom. 2 (1968), 77-159. MR 0236328 (38:4625a)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C30, 53C35

Retrieve articles in all journals with MSC: 53C30, 53C35


Additional Information

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

American Mathematical Society