Curvature homogeneous hypersurfaces immersed in a real space form
type: Item
instance of: scholarly article
title: Curvature homogeneous hypersurfaces immersed in a real space form
zbMATH Open document ID: 0651.53037
author: K. Tsukada
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1988-01-01T00:00:00Z
review text: 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. \textit{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 \(H^ 5(-1)\).
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 4062094
zbMATH Keywords: classification
zbMATH Keywords: curvature homogeneous space
zbMATH Keywords: immersed hypersurface
zbMATH Keywords: real space form
zbMATH Keywords: constant principal curvatures
reviewed by: Kouei Sekigawa
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228028
OpenAlex ID: W1977663541
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DOI: 10.2748/TMJ/1178228028
description: scientific article; zbMATH DE number 4062094