Metrics with harmonic curvature.

*(English)*Zbl 0622.53026
Global Riemannian geometry, Proc. Symp., Durham/Engl. 1982, 18-26 (1984).

[For the entire collection see Zbl 0614.00017.]

The paper is a survey of most important results on compact manifolds with harmonic curvature known by 1982. In particular, the author discusses two different proofs of his theorem [Invent. Math. 63, 263-286 (1981; Zbl 0456.53033)], stating that a manifold of dimension four with nonzero signature must be Einstein.

(Since the paper was written, further results in this direction were obtained, a discussion of which can be found in Ch. 16 of ”Einstein manifolds” by A. L. Besse (Springer 1987; Zbl 0613.53001); see also the reviewer’s survey paper, Riemannian manifolds with harmonic curvature [Lect. Notes Math. 1156, pp. 74-85 (1985; Zbl 0574.53032)]. The author’s question whether there exist compact manifolds with harmonic curvature that are simply connected without being products of conformally flat and/or Einstein manifolds, has been given an affirmative answer by the reviewer [Riemannian metrics with harmonic curvature on 2-sphere bundles over compact surfaces to appear in Bull. Soc. Math. Fr. 116, 111-134 (1988)].)

The paper is a survey of most important results on compact manifolds with harmonic curvature known by 1982. In particular, the author discusses two different proofs of his theorem [Invent. Math. 63, 263-286 (1981; Zbl 0456.53033)], stating that a manifold of dimension four with nonzero signature must be Einstein.

(Since the paper was written, further results in this direction were obtained, a discussion of which can be found in Ch. 16 of ”Einstein manifolds” by A. L. Besse (Springer 1987; Zbl 0613.53001); see also the reviewer’s survey paper, Riemannian manifolds with harmonic curvature [Lect. Notes Math. 1156, pp. 74-85 (1985; Zbl 0574.53032)]. The author’s question whether there exist compact manifolds with harmonic curvature that are simply connected without being products of conformally flat and/or Einstein manifolds, has been given an affirmative answer by the reviewer [Riemannian metrics with harmonic curvature on 2-sphere bundles over compact surfaces to appear in Bull. Soc. Math. Fr. 116, 111-134 (1988)].)

Reviewer: A.Derdzinski