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 determined, 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.