Three-dimensional curvature homogeneous hypersurfaces. (English) Zbl 1054.53070

The study of curvature homogeneous Riemannian manifolds (as generalizations of locally homogeneous spaces) was initiated by I. M. Singer in 1960 and intensively developed by many authors in recent years. K. Tsukada solved completely the problem which curvature homogeneous spaces of dimension \(n>3\) can be isometrically immersed as hypersurfaces of space forms. The case \(n=3\) remains an interesting and rather difficult open problem. A conjecture says that all these spaces are locally homogeneous. The present authors derive various proofs of this conjecture under additional conditions. Such conditions are e.g. compactness or various inequalities for the curvature.


53C30 Differential geometry of homogeneous manifolds
53C20 Global Riemannian geometry, including pinching
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