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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
