Cohomologically Calibrated Affine Connections and Forced Irreducibility

TL;DR

This paper proves that certain affine connections on product manifolds are necessarily irreducible when their calibration class is mixed, using Hodge theory and integral arguments, with implications for geometric structures and quantum analogies.

Contribution

It establishes a new principle of forced geometric irreducibility for cohomologically calibrated affine connections on product manifolds, expanding understanding of their holonomy properties.

Findings

Cohomologically calibrated affine connections are necessarily irreducible on product manifolds.

The proof uses Hodge theory to analyze harmonic torsion components.

Explicit examples on $S^2\times \Sigma_g$ demonstrate the theorem's applicability.

Abstract

We establish a principle of forced geometric irreducibility on product manifolds. We prove that for any product manifold $M=M_1\times M_2$, a cohomologically calibrated affine connection, $\nabla^{\mathcal{C}}$, is necessarily holonomically irreducible, provided its calibration class $[\omega] \in H^3(M;\mathbb{R})$ is mixed. The core of the proof relies on Hodge theory; we show that the algebraic structure of the harmonic part of the torsion generates non-zero off-diagonal components in the full Riemann curvature tensor, which cannot be globally cancelled. This non-cancellation is formally proven via an integral argument. We illustrate the main theorem with explicit constructions on $S^2\times \Sigma_g$, showing that this result holds even in special cases where the Ricci tensor is diagonal, such as the Einstein-calibrated connection. Finally, we briefly discuss speculative analogies between forced irreducibility and quantum entanglement.