Harmonic curvature in dimension four
Andrzej Derdzinski
TL;DR
This work advances the classification of four-dimensional Riemannian manifolds with harmonic curvature by showing that, outside the known local types, generic points admit a Ricci tensor with four distinct eigenvalues and a local frame in which $g$, $\mathrm{Ric}$, and $R$ are simultaneously diagonalizable. It further proves that the 12 connection coefficients $F_{ji}^{\,i}$ and the 6 curvature components $R_{ijij}^{\,i}$ obey a fixed, pointwise polynomial system, so the local geometry is constrained to a real algebraic variety. The paper develops a framework connecting differential geometry with real algebraic geometry through a detailed study of algebraic Weyl tensors, Codazzi diagonalizability, and the associated a priori case analysis, including the treatment of compact-type examples (kne e) and the explicit local-type classification (kne a–d). The results lay groundwork for identifying new examples or ruling out possibilities by translating geometric questions into algebraic ones, with potential implications for the structure of generic harmonic-curvature four-manifolds. Overall, the work provides a rigorous path from differential-geometric conditions to real-algebraic constraints, clarifying the landscape of four-manifolds with harmonic curvature and highlighting the role of diagonalizability, eigenvalue structures, and warped-product phenomena in their local geometry.
Abstract
We provide a step towards classifying Riemannian four-manifolds 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.