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Curvature homogeneous hypersurfaces immersed in a real space form. (English) Zbl 0651.53037
A curvature homogeneous space is a Riemannian manifold M satisfying the condition: for each \(x,y\in M\), there exists a linear isometry \(\Phi\) of \(T_ xM\) onto \(T_ yM\) such that \(\Phi (R_ x)=R_ y\), where R is the curvature tensor of M. H. Takagi gave the first example of a simply connected complete curvature homogeneous space which is not homogeneous [ibid. 26, 581-585 (1974; Zbl 0302.53022)]. An immersed hypersurface in a real space form with constant principal curvatures is curvature homogeneous. In the present paper, the author classifies curvature homogeneous hypersurfaces in an \((n+1)\)-dimensional Euclidean space \(E^{n+1}\) (n\(\geq 3)\), an \((n+1)\)-dimensional unit sphere \(S^{n+1}(1)\) (n\(\geq 4)\) and an \((n+1)\)-dimensional hyperbolic space \(H^{n+1}(-1)\) (n\(\geq 4)\) of constant curvature -1. In the course of the classification, the author constructs an example of complete curvature homogeneous hypersurface of cohomogeneity one in \(H^ 5(-1)\).
Reviewer: K.Sekigawa

MSC:
53C40 Global submanifolds
53C20 Global Riemannian geometry, including pinching
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[1] M. DAJCZER AND D. GROMOLL, Gauss parametrizations and rigidity aspects of submani-folds, J. Differential Geometry 22 (1985), 1-12. · Zbl 0588.53007
[2] D. FERUS, Totally geodesic foliations, Math. Ann. 188 (1970), 313-316 · Zbl 0194.52804
[3] D. FERUS, H. KARCHER AND H. F. MUNZNER, Cliffordalgebren und neue isoparametrisch Hyperflachen, Math. Z. 177 (1981), 479-502. · Zbl 0443.53037
[4] S. KOBAYASHI AND K. NOMIZU, Foundations of Differential Geometry, Vol. 2, Interscience, New York, London, Sydney, 1969. · Zbl 0526.53001
[5] J. D. MOORE, Isometric immersions of Riemannian products, J. Differential Geometry (1971), 159-168. · Zbl 0213.23804
[6] T. NAGANO AND T. TAKAHASHI, Homogeneous hypersurfaces in Euclidean spaces, J. Math Soc.Japan 12 (1960), 1-7. · Zbl 0102.16403
[7] H. NAKAGAWA, Riemannian geometry in the large (in Japanese), Kaigai Publicatio L. T. D. 1977. · Zbl 0361.90027
[8] H. OZEKI AND M. TAKEUCHI, On some types of isoparametric hypersurfaces in spheres, I, II, Thoku Math. J. 27 (1975), 515-559; ibid. 28 (1976), 7-55. · Zbl 0359.53011
[9] P. J. RYAN, Homogeneity and some curvature conditions for hypersurfaces, Thok Math. J. 21 (1969), 363-388. · Zbl 0185.49904
[10] K. SEKIGAWA, On some 3-dimensional Riemannian manifolds, Hokkaido Math. J. 2 (1973), 259-270. · Zbl 0266.53034
[11] I. M. SINGER, Infinitesimally homogeneous spaces, Comm. Pure Appl. Math. 13 (1960), 685-697. · Zbl 0171.42503
[12] Z. I. SZABO, Structure theorems on Riemannian spaces satisfying R(X, Y)-R = 0, I, th local version, J. Differential Geometry 17 (1982), 531-582; II, global versions, Geo-metriae Dedicata 19 (1985), 65-108. · Zbl 0508.53025
[13] H. TAKAGI, On curvature homogeneity of Riemannian manifolds, Thoku Math. J. 2 (1974), 581-585. · Zbl 0302.53022
[14] T. TAKAHASHI, Homogeneous hypersurf aces in spaces of constant curvature, J. Math Soc. Japan 22 (1970), 395-410. · Zbl 0189.22501
[15] T. TAKAHASHI, An isometric immersion of a homogeneous Riemannian manifold of di mension 3 in the hyperbolic space, J. Math. Soc. Japan 23 (1971), 649-661. · Zbl 0218.53078
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