Positive biorthogonal curvature on $S^2 \times T^2$ via affine connection
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911061694742528 |
|---|---|
| author | Pigazzini, Alexander |
| author_facet | Pigazzini, Alexander |
| contents | We address the long-standing problem of the existence of a Riemannian metric on \(S^2\times T^2\) with strictly positive biorthogonal curvature (\( K_{\text{biort}}(σ) > 0 \)). This work tackles this challenge within a weaker, yet geometrically consistent, framework by introducing an affine connection, topologically motivated, on \( S^2 \times T^2 \) with antisymmetric torsion. Crucially, this torsion is calibrated via non-trivial cohomology classes in \( H^3(S^2 \times T^2; \mathbb{R}) \cong \mathbb{R}^2 \), an approach that allows overcoming topological constraints such as \( χ= 0 \). We demonstrate that this construction, while not requiring metric compatibility (though retaining the metric ( \(g\) ) for norms and orthogonality), successfully yields strictly positive biorthogonal curvature across the manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_11914 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Positive biorthogonal curvature on $S^2 \times T^2$ via affine connection Pigazzini, Alexander Differential Geometry 53C05, 53C20, 53C21 We address the long-standing problem of the existence of a Riemannian metric on \(S^2\times T^2\) with strictly positive biorthogonal curvature (\( K_{\text{biort}}(σ) > 0 \)). This work tackles this challenge within a weaker, yet geometrically consistent, framework by introducing an affine connection, topologically motivated, on \( S^2 \times T^2 \) with antisymmetric torsion. Crucially, this torsion is calibrated via non-trivial cohomology classes in \( H^3(S^2 \times T^2; \mathbb{R}) \cong \mathbb{R}^2 \), an approach that allows overcoming topological constraints such as \( χ= 0 \). We demonstrate that this construction, while not requiring metric compatibility (though retaining the metric ( \(g\) ) for norms and orthogonality), successfully yields strictly positive biorthogonal curvature across the manifold. |
| title | Positive biorthogonal curvature on $S^2 \times T^2$ via affine connection |
| topic | Differential Geometry 53C05, 53C20, 53C21 |
| url | https://arxiv.org/abs/2502.11914 |