Cohomological Calibration and Curvature Constraints on Product Manifolds: A Topological Lower Bound

Abstract: We establish a quantitative relationship between mixed cohomology classes and the geometric complexity of cohomologically calibrated metric connections with totally skew torsion on product manifolds. Extending the results of Pigazzini--Toda (2025), we show that the dimension of the off-diagonal curvature subspace of a connection $\nabla^C$ is bounded below by the sum of tensor ranks of the mixed K\"unneth components of its calibration class. The bound depends only on the mixed class $[\omega]_{\mathrm{mixed}}\in H^3(M;\mathbb{R})$, hence is topological and independent of the chosen product metric. This provides a computational criterion for geometric complexity and quantifies the interaction between topology and curvature, yielding a quantified version of ``forced irreducibility'' via the dimension of $\mathfrak{hol}_p^{\mathrm{off}}(\nabla^C)$.