Curvature homogeneous hypersurfaces immersed in a real space form. (English) Zbl 0651.53037

Tsukada, Kazumi

Tôhoku Math. J., II. Ser. 40, No. 2, 221-244 (1988).

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

Keywords:

classification; curvature homogeneous space; immersed hypersurface; real space form; constant principal curvatures

Citations:

Zbl 0302.53022

References:

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