Classification of certain compact Riemannian manifolds with harmonic curvature and non-parallel Ricci tensor

Abstract

S being the Ricci tensor. While every manifold with parallel Ricci tensor has harmonic curvature, i.e., satisfies fiR=O, there are examples ([3], Theorem 5.2) of open Riemannian manifolds with fiR=O and VS+O. In [1] Bourguignon has asked the question whether the Ricci tensor of a compact Riemannian manifold with harmonic curvature must be parallel. The aim of this paper is to give examples (see Remark 2) answering this question in the negative. All our examples are conformally flat (Corollary 1). Moreover, we obtain some classification results, restricting our consideration to Riemannian manifolds with fiR = O, VS + 0 and such that the Ricci tensor S has at any point less than three distinct eigenvalues. Starting from a description of their local structure at generic points (Theorem 1), we find all four-dimensional, analytic, complete and simply connected manifolds of this type (Theorem 2). They are all non-compact, but some of them do possess compact quotients. Next we prove (Theorem 3) that all compact four-dimensional analytic Riemannian manifolds with the above properties are covered by S 1 x S 3 with a metric of an explicitly described form. Throughout this paper, by a manifold we mean a connected paracompact manifold of class C ~ or analytic. By abuse of notation, concerning Riemannian manifolds we often write M instead of (M,g) and @ , v ) instead of g(u,v) for tangent vectors u, v.