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Classification of simply connected four-dimensional $ RR$-manifolds


Authors: Gr. Tsagas and A. Ledger
Journal: Trans. Amer. Math. Soc. 219 (1976), 189-210
MSC: Primary 53C30
DOI: https://doi.org/10.1090/S0002-9947-1976-0467603-0
MathSciNet review: 0467603
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Abstract: Let (M, g) be a Riemannian manifold. We assume that there is a mapping $ s:M \to I(M)$, where $ I(M)$ is the group of isometries of (M, g), such that $ {s_x} = s(x),\forall x \in M$, has x as a fixed isolated point, then (M, g) is called a Riemannian s-manifold. If the tensor field S on M defined by the relation $ {S_x} = {(d{s_x})_x},\forall x \in M$, is differentiable and invariant by each isometry $ {s_x}$, then the manifold (M, g) is called a regularly s-symmetric Riemannian manifold. The aim of the present paper is to classify simply connected four-dimensional regularly s-symmetric Riemannian manifolds.


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  • [1] P. J. Graham and A. J. Ledger, s-regular manifolds, Differential Geometry in Honor of Kentaro Yono, Tokyo, 1972, pp. 133-144. MR 0328825 (48:7167)
  • [2] A. Gray, Riemannian manifolds with geodesic symmetries of order 3, J. Differential Geometry 7 (1972), 343-369. MR 0331281 (48:9615)
  • [3] S. Helgason, Differential geometry and symmetric spaces, Pure and Appl. Math., vol. 12, Academic Press, New York and London, 1962. MR 26 #2986. MR 0145455 (26:2986)
  • [4] S. Ishihara, Homogeneous Riemannian spaces of four dimensions, J. Math. Soc. Japan 7 (1955), 345-370. MR 18, 599. MR 0082717 (18:599b)
  • [5] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vols. 1, 2, Interscience, New York and London, 1963, 1969. MR 27 #2945; 38 #6501.
  • [6] D. Kowalki and A. J. Ledger, Regular s-structures on manifolds (to appear).
  • [7] A. J. Ledger and M. Obata, Affine and Riemannian s-manifolds, J. Differential Geometry 2 (1968), 451-459. MR 39 #6206. MR 0244893 (39:6206)
  • [8] K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33-65. MR 15, 468. MR 0059050 (15:468f)
  • [9] J. A. Wolf, Spaces of constant curvature, McGraw-Hill, New York, 1967. MR 36 #829. MR 0217740 (36:829)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0467603-0
Keywords: Simply connected manifold, Lie group, Lie algebra, s-structure, Riemannian s-manifold, symmetry tensor field, symmetric space, adjoint group
Article copyright: © Copyright 1976 American Mathematical Society

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