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

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.