Title: Compact riemannian manifolds with harmonic curvature and non-parallel ricci tensor
Abstract: While every manifold with parallel Ricci tensor has harmonic curvature (i.e. satisfies 6R = 0), there are examples ([3], Theorem 5.2) of open Riemannian manifolds with 6R = 0 and VS ~ O. In [1] J.P. Bourguignon has asked the question whether the Ricci tensor of a compact Riemannian manifold with harmonic curvature must be parallel. The aim of this note is to describe an easy example answering this question in the negative. More precisely, metrics with 6R = 0 and VS ~ 0 are exhibited on S 1 × N 3, N 3 being e.g. the 3-sphere or a lens space. By taking products of these manifolds with themselves or with arbitrary compact Einstein manifolds, one gets similar examples in all dimensions greater {han three.
Publication Year: 1981
Publication Date: 1981-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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Cited By Count: 18
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