Cohomological Calibration and Curvature Constraints on Product Manifolds: A Topological Lower Bound
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913041304518656 |
|---|---|
| author | Pigazzini, Alexander Toda, Magdalena |
| author_facet | Pigazzini, Alexander Toda, Magdalena |
| contents | We establish a quantitative relationship between mixed de Rham classes and the geometric complexity of metric connections with totally skew torsion on product manifolds where both factors are compact oriented surfaces. For any cohomologically calibrated connection $\nabla^C$ whose torsion $T$ has pure bidegree with respect to the product decomposition and whose harmonic projection represents a non-trivial mixed class $[ω]$, we prove that on a non-empty open subset $\mathcal{V} \subset M$, \[ \dim\bigl(\mathfrak{hol}_p^{\mathrm{off}}(\nabla^{C})\bigr)\;\geq\; r^\sharp\;:=\;\operatorname{rank}_{\mathbb{R}}\bigl([ω]_{\mathrm{mixed}}\bigr)-\dim\mathcal{K}, \] with $\mathcal{K}$ an intrinsically defined obstruction space. The bound is a topological invariant under metric deformations preserving the parallel-form strata and provides an obstruction to reducible holonomy. Counterexamples show the hypothesis is optimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11834 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohomological Calibration and Curvature Constraints on Product Manifolds: A Topological Lower Bound Pigazzini, Alexander Toda, Magdalena Differential Geometry Algebraic Topology 53C05, 53C07, 53C29, 58A14, 57R19, 15A69 We establish a quantitative relationship between mixed de Rham classes and the geometric complexity of metric connections with totally skew torsion on product manifolds where both factors are compact oriented surfaces. For any cohomologically calibrated connection $\nabla^C$ whose torsion $T$ has pure bidegree with respect to the product decomposition and whose harmonic projection represents a non-trivial mixed class $[ω]$, we prove that on a non-empty open subset $\mathcal{V} \subset M$, \[ \dim\bigl(\mathfrak{hol}_p^{\mathrm{off}}(\nabla^{C})\bigr)\;\geq\; r^\sharp\;:=\;\operatorname{rank}_{\mathbb{R}}\bigl([ω]_{\mathrm{mixed}}\bigr)-\dim\mathcal{K}, \] with $\mathcal{K}$ an intrinsically defined obstruction space. The bound is a topological invariant under metric deformations preserving the parallel-form strata and provides an obstruction to reducible holonomy. Counterexamples show the hypothesis is optimal. |
| title | Cohomological Calibration and Curvature Constraints on Product Manifolds: A Topological Lower Bound |
| topic | Differential Geometry Algebraic Topology 53C05, 53C07, 53C29, 58A14, 57R19, 15A69 |
| url | https://arxiv.org/abs/2509.11834 |