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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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##### References:
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