Hodge Splittings and Einstein 4-manifolds

Summary · qwen2.5:32b

The study explores pairs of Riemannian metrics $(g,h)$ on oriented 4-manifolds where $g$'s curvature tensor preserves $h$'s Hodge splitting, extending the Einstein condition to a broader class when $h$ is conformal to $g$. This extension satisfies a generalized Hitchin-Thorpe inequality that collapses to the classical form under conformality. A specific example on $\#_5\mathbb{CP}^2$ violates the Hitchin-Thorpe inequality, demonstrating the existence of manifolds admitting metrics satisfying this broader condition but not Einstein metrics.

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Excerpt

arXiv:2508.08118v3 Announce Type: replace Abstract: On an oriented $4$-manifold, we study pairs of Riemannian metrics $(g,h)$ for which the curvature tensor of $g$ preserves the Hodge splitting determined by $h$. This extends the Einstein condition in dimension four, which is recovered when $h$ is conformal to $g$. We prove that such pairs satisfy a generalized Hitchin-Thorpe inequality, which reduces to the classical one when $h$ is conformal to $g$. We then exhibit a pair $(g,h)$ on $\#_5\mathbb{CP}^2$, which violates Hitchin-Thorpe and hence admits no Einstein metric, thus showing that our condition is indeed broader than the Einstein condition.