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

Abstract

We establish a quantitative relationship between mixed de Rham classes and the geometric complexity of metric connections with totally skew torsion on product manifolds where both factors are compact oriented surfaces. For any cohomologically calibrated connection $\nabla^C$ whose torsion $T$ has pure bidegree with respect to the product decomposition and whose harmonic projection represents a non-trivial mixed class $[\omega]$, we prove that on a non-empty open subset $\mathcal{V} \subset M$, \[ \dim\bigl(\mathfrak{hol}_p^{\mathrm{off}}(\nabla^{C})\bigr)\;\geq\; r^\sharp\;:=\;\operatorname{rank}_{\mathbb{R}}\bigl([\omega]_{\mathrm{mixed}}\bigr)-\dim\mathcal{K}, \] with $\mathcal{K}$ an intrinsically defined obstruction space. The bound is a topological invariant under metric deformations preserving the parallel-form strata and provides an obstruction to the reduction of the holonomy along the product splitting $V_1\oplus V_2$. A counterexample shows the hypothesis is optimal. When the second factor contains a circle factor, we further show that $r^{\sharp}=1$ forces the torsion to survive dimensional reduction along it, so that the failure of the holonomy to preserve the product splitting persists on the reduced product; the mixed rank alone cannot detect this.