Journal of the
Korean Mathematical Society JKMS

We provide a step towards classifying Riemannian four-man\-i\-folds in which the curvature tensor has zero divergence, or -- equivalently -- the Ricci tensor Ric satisfies the Codazzi equation. Every known compact manifold of this type belongs to one of five otherwise-familiar classes of examples. The main result consists in showing that, if such a manifold (not necessarily compact or even complete) lies outside of the five classes -- a non-vacuous assumption -- then, at all points of a dense open subset, Ric has four distinct eigenvalues, while suitable local coordinates simultaneously diagonalize Ric, the metric and, in a natural sense, also the curvature tensor. Furthermore, in a local orthonormal frame formed by Ricci eigenvectors, the connection form (or, curvature tensor) has just twelve (or, respectively, six) possibly-nonzero components, which together satisfy a specific system, not depending on the point, of homogeneous polynomial equations. A part of the classification problem is thus reduced to a question in real algebraic geometry.

Keywords: Harmonic curvature, Co\-daz\-zi tensor

MSC numbers: Primary 53B20; Secondary 53C25

Supported by: Research supported in part by a FAPESP\hn-\hh OSU 2015 Regular Research Award (FAPESP grant: 2015/50265-6).