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New explicit examples of complete semi-symmetric hypersurfaces of hyperbolic type. (English) Zbl 0885.53006
A Riemannian manifold \((M,g)\) is said to be semi-symmetric if, at each point \(p\in M\), the curvature tensor is the same as that of a symmetric space, which may vary with the point \(p\). This class of spaces has been treated extensively, and from the local and global viewpoint, by Z. I. Szabó. The explicit description of the metrics has been considered in full detail in [E. Boeckx, O. Kowalski and L. Vanhecke, ‘Riemannian manifolds of conullity two’ (World Scientific Singapore) (1996)].
Complete semi-symmetric hypersurfaces in \(\mathbb{R}^{n+1}\) have been studied by K. Nomizu, R. Takagi and others, in particular by Z. I. Szabó. He considered several special subclasses, in particular the so-called complete semi-symmetric spaces of hyperbolic type. This class contains the example constructed by Takagi in \(\mathbb{R}^4\) as a special case. This example was the only explicitly known non-symmetric one. In this paper, the author constructs, using holomorphic functions and the general construction procedure developed by Szabó, a large family of new examples in \(\mathbb{R}^4\). He also shows that the restriction to dimension four is not important since, for higher dimensions and in the simply connected case, one only obtains Riemannian products \(M^3\times\mathbb{R}^{n-3}\) in \(\mathbb{R}^{n+1}\).
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53C35 Differential geometry of symmetric spaces
53C20 Global Riemannian geometry, including pinching
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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