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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2508.08118 |
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| _version_ | 1866908981447884800 |
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| author | Aazami, Amir Babak |
| author_facet | Aazami, Amir Babak |
| contents | 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 $h = g$. We show that this extension admits a variational characterization: for fixed $g$, the admissible auxiliary metrics $h$ are precisely the critical points of the conformally invariant mixed Einstein-Hilbert functional $\int_M \text{scal}_{\text{$g$-$h$}} dV_h$, where $\text{scal}_{\text{$g$-$h$}}$ is the $h$-scalar contraction of the curvature tensor of $g$. We also compute the second variation and show that pointwise nondegeneracy of the induced Hessian on trace-free symmetric 2-tensors yields local rigidity and persistence of admissible conformal classes under perturbations of $g$. Finally, we exhibit non-Einstein examples of $(g, h)$ on products of surfaces and on $\mathbb{S}^4$, and, under a shared-orthogonal-frame ansatz, obtain a Berger-type nonnegativity result for the Euler characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08118 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hodge Splittings and Einstein 4-manifolds Aazami, Amir Babak Differential Geometry 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 $h = g$. We show that this extension admits a variational characterization: for fixed $g$, the admissible auxiliary metrics $h$ are precisely the critical points of the conformally invariant mixed Einstein-Hilbert functional $\int_M \text{scal}_{\text{$g$-$h$}} dV_h$, where $\text{scal}_{\text{$g$-$h$}}$ is the $h$-scalar contraction of the curvature tensor of $g$. We also compute the second variation and show that pointwise nondegeneracy of the induced Hessian on trace-free symmetric 2-tensors yields local rigidity and persistence of admissible conformal classes under perturbations of $g$. Finally, we exhibit non-Einstein examples of $(g, h)$ on products of surfaces and on $\mathbb{S}^4$, and, under a shared-orthogonal-frame ansatz, obtain a Berger-type nonnegativity result for the Euler characteristic. |
| title | Hodge Splittings and Einstein 4-manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2508.08118 |