Weakly Einstein Kahler Surfaces in Complex Differential Geometry
A weakly Einstein Riemannian four-manifold is one whose triple (three-index) self-contraction of the curvature tensor equals a function times the metric; in dimension four every Einstein manifold has this property. This work studies the Kahler case, giving several conditions equivalent to a Kahler surface being weakly Einstein, classifying weakly Einstein Kahler surfaces under additional hypotheses (locally homogeneous, self-dual, or compact non-Einstein), and constructing new examples.
2501.00649
A Riemannian four-manifold is called weakly Einstein when the three-index contraction of its curvature tensor with itself equals a function times the metric, a condition satisfied in dimension four b…