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Zbl 0681.53020
Derdziński, Andrzej
Riemannian metric with harmonic curvature on 2-sphere bundles over compact surfaces. (English)
[J] Bull. Soc. Math. Fr. 116, No.2, 133-156 (1988). ISSN 0037-9484

The author exhibits examples of continuous families of Riemannian metrics on $S\sp 2$-bundles over surfaces with harmonic curvature which are neither conformally flat with constant scalar curvature nor Einstein. They are obtained by imposing the extra condition that the metrics be locally conformal to a product of metrics (this forces two eigenvalues of the Weyl tensor to be equal), and solving a nonlinear partial differential equation satisfied by the Gaussian curvature of the factors using a bifurcation argument. A geometric characterization of these metrics is also given.
[J.P.Bourguignon]

MSC 2000:: *53C20 Riemannian manifolds (global)

Keywords: $S\sp 2$-bundles; surfaces; harmonic curvature; Weyl tensor; Gaussian curvature

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