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Riemannian manifolds whose curvature operator R(X, Y) has constant eigenvalues

Published online by Cambridge University Press:  17 April 2009

Y. Nikolayevsky
Affiliation:
Department of Mathematics, La Trobe University, Bundoora, Vic 3086, Australia e-mail: Y.Nikolayevsky@latrobe.edu.au
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A Riemannian manifold Mn is called IP, if, at every point xMn, the eigenvalues of its skew-symmetric curvature operator R(X, Y) are the same, for every pair of orthonormal vectors X, YTxMn. In [5, 6, 12] it was shown that for all n ≥ 4, except n = 7, an IP manifold either has constant curvature, or is a warped product, with some specific function, of an interval and a space of constant curvature. We prove that the same result is still valid in the last remaining case n = 7, and also study 3-dimensional IP manifolds.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2004

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