Cohomologically calibrated affine connections and the Einstein condition on $S^2 \times T^2$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912518578896896 |
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| author | Pigazzini, Alexander Toda, Magdalena |
| author_facet | Pigazzini, Alexander Toda, Magdalena |
| contents | This paper applies the recently developed framework of cohomologically calibrated affine connections to the fundamental problem of constructing non-Riemannian Einstein manifolds. In this framework, the torsion of a connection is intrinsically related to the global topology of the manifold, represented by the de Rham cohomology class specified by a set of real parameters. We focus on the product manifold $S^2 \times T^2$, whose third cohomology group is $H^3(S^2 \times T^2; \mathbb{R}) \cong \mathbb{R}^2$. We analyze how the geometry is modeled by the choice of the torsion tensor $T$ within the family $\mathcal{T}_ω$, defined by the property that each member of this family must have an associated 3-form $T^\flat$ such that it represents the nontrivial cohomology class via Hodge decomposition. Our analysis reveals a dependence on this choice. First, we show that using a torsion tensor, which produces a strictly positive biorthogonal curvature, leads to a non-diagonal Ricci tensor, creating a structural obstacle to any Einstein solution. Conversely, we then show that using the torsion tensor associated with the purelly harmonic 3-form allow us the construction of an explicit non-Riemannian Einstein solution. Our work thus demonstrates that the cohomologically calibrated affine connections allow a family of feasible connections rich enough to allow for several geometries intrinsically justified by the differential topology of the manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02353 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohomologically calibrated affine connections and the Einstein condition on $S^2 \times T^2$ Pigazzini, Alexander Toda, Magdalena Differential Geometry 53C05, 53C15, 53C25, 58A14 This paper applies the recently developed framework of cohomologically calibrated affine connections to the fundamental problem of constructing non-Riemannian Einstein manifolds. In this framework, the torsion of a connection is intrinsically related to the global topology of the manifold, represented by the de Rham cohomology class specified by a set of real parameters. We focus on the product manifold $S^2 \times T^2$, whose third cohomology group is $H^3(S^2 \times T^2; \mathbb{R}) \cong \mathbb{R}^2$. We analyze how the geometry is modeled by the choice of the torsion tensor $T$ within the family $\mathcal{T}_ω$, defined by the property that each member of this family must have an associated 3-form $T^\flat$ such that it represents the nontrivial cohomology class via Hodge decomposition. Our analysis reveals a dependence on this choice. First, we show that using a torsion tensor, which produces a strictly positive biorthogonal curvature, leads to a non-diagonal Ricci tensor, creating a structural obstacle to any Einstein solution. Conversely, we then show that using the torsion tensor associated with the purelly harmonic 3-form allow us the construction of an explicit non-Riemannian Einstein solution. Our work thus demonstrates that the cohomologically calibrated affine connections allow a family of feasible connections rich enough to allow for several geometries intrinsically justified by the differential topology of the manifold. |
| title | Cohomologically calibrated affine connections and the Einstein condition on $S^2 \times T^2$ |
| topic | Differential Geometry 53C05, 53C15, 53C25, 58A14 |
| url | https://arxiv.org/abs/2508.02353 |