Positive biorthogonal curvature on $S^2 \times T^2$ via affine connection

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 (\( K_{\text{biort}}(\sigma) > 0 \)). 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 \( \chi = 0 \). 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.