Classification of simply connected four-dimensional $RR$-manifolds
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- by Gr. Tsagas and A. Ledger PDF
- Trans. Amer. Math. Soc. 219 (1976), 189-210 Request permission
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.References
- P. J. Graham and A. J. Ledger, $s$-regular manifolds, Differential geometry (in honor of Kentaro Yano), Kinokuniya, Tokyo, 1972, pp. 133–144. MR 0328825
- Alfred Gray, Riemannian manifolds with geodesic symmetries of order $3$, J. Differential Geometry 7 (1972), 343–369. MR 331281
- Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
- Shigeru Ishihara, Homogeneous Riemannian spaces of four dimensions, J. Math. Soc. Japan 7 (1955), 345–370. MR 82717, DOI 10.2969/jmsj/00740345 S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vols. 1, 2, Interscience, New York and London, 1963, 1969. MR 27 #2945; 38 #6501. D. Kowalki and A. J. Ledger, Regular s-structures on manifolds (to appear).
- A. J. Ledger and M. Obata, Affine and Riemannian $s$-manifolds, J. Differential Geometry 2 (1968), 451–459. MR 244893, DOI 10.4310/jdg/1214428660
- Katsumi Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33–65. MR 59050, DOI 10.2307/2372398
- Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York-London-Sydney, 1967. MR 0217740
Additional Information
- © Copyright 1976 American Mathematical Society
- 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