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 $\nabla^C$ with non-parallel torsion $T$, where the harmonic projection of $T$ represents a mixed class $[\omega]$, we prove that on a non-empty open subset $\mathcal{ V} \subset M$, $ \dim\mathfrak{ hol}_p^{ \mathrm{ off}}(\nabla^{ C})\ge \min\bigl(\operatorname{ rank}_{ \mathbb{ R}}([\omega]_{ \mathrm{ mixed}})-\dim\mathcal{ K},\;1\bigr), $ 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 reducible holonomy. Counterexamples show the hypothesis is optimal.
Categories: math.DG math.AT