Year: 2025
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Aazami, Amir Babak
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416243382
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