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Riemannian manifold in which the skew-symmetric curvature operator has pointwise constant eigenvalues. (English) Zbl 0903.53016

Let \((M,g)\) be a Riemannian manifold and denote by \(R\) its associated Riemannian curvature tensor. Let \(p\) be a point of \(M\) and consider the two-dimensional subspace of \(T_pM\) determined by the two orthonormal tangent vectors \(u, v\). Then the curvature operator \(R_{uv}\) is skew-symmetric and depends only on the two-plane and the orientation of the orthonormal basis \((u,v)\).
In this paper, the authors derive a classification of the four-dimensional manifolds for which the eigenvalues of this operator are independent of the two-plane at \(p\), i.e., are pointwise constant. They show that such manifolds are almost everywhere locally isometric to a real space form or a warped product \(I\times_f N\) where \(I\) is an open interval of \(\mathbb{R}\), \(N\) is a space of constant curvature \(k\) and \(f^2= kt^2+ Ct+ D\), \(t\in I\), where \(C\) and \(D\) are constants such that \(C^2- 4kD\neq 0\). Moreover, the eigenvalues are globally constant on a connected manifold \(M\) if and only if \((M,g)\) is a real space form.
The authors also treat the three-dimensional case. Here the results are different and a complete classification is not yet known. P. B. Gilkey, J. V. Leahy and H. Sadofsky [‘Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues’ (Preprint; http://hopf.uoregon.edu/\(\sim\)gilkey/dirresearch/prepap.html)] announced a generalization of the four-dimensional results to all other dimensions except for dimensions 7 and 8.

MSC:

53B20 Local Riemannian geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53B35 Local differential geometry of Hermitian and Kählerian structures
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