Preprint
Inserted: 16 sep 2025
Last Updated: 14 apr 2026
Year: 2025
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 reducible holonomy. Counterexamples show the hypothesis is optimal.
Keywords: Curvature Tensor, Künneth Decomposition, Hodge Theory, De Rham Cohomology, Holonomy Algebra, Product Manifolds, Connections with Torsion, PT lower bound
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