Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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HARMONIC CURVATURE IN DIMENSION FOUR

  • Received : 2024.01.01
  • Accepted : 2024.08.14
  • Published : 2025.01.01
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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.

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Acknowledgement

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

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