preprint
Inserted: 15 apr 2025
Last Updated: 15 apr 2025
Pages: 13
Year: 2025
Abstract:
We address the long-standing problem of the existence of a Riemannian metric on \(S^2\times T^2\) with strictly positive biorthogonal curvature (\( K_{\text{biort}}(\sigma) > 0 \)), but in a weaker framework, by introducing an affine connection with antisymmetric closed torsion, naturally encoded in the cohomology of \(S^2 \times T^2\) (\(H^3(S^2 \times T^2; \mathbb{R}) \cong \mathbb{R}^2\)). This torsion, parametrized by non-trivial cohomology classes, overcomes topological constraints imposed by the zero Euler characteristic, ensuring \( K_{\text{biort}}(\sigma) > 0 \) globally.
Keywords: Positive biorthogonal curvature, 4-manifolds, non-simply connected manifold, affine connection, torsion