Cohomologically calibrated affine connections and the Einstein condition on $S^2 \times T^2$

TL;DR

This paper explores the use of cohomologically calibrated affine connections on the manifold $S^2 imes T^2$ to construct non-Riemannian Einstein manifolds, linking topology with geometric structures.

Contribution

It introduces a framework connecting torsion tensors to cohomology classes, enabling the construction of explicit non-Riemannian Einstein solutions on $S^2 imes T^2$.

Findings

Positive biorthogonal curvature torsion leads to non-Einstein Ricci tensors.

Harmonic 3-form torsion allows explicit Einstein solutions.

Topology influences the feasibility of Einstein geometries.

Abstract

This paper applies the recently developed framework of cohomologically calibrated affine connections to the fundamental problem of constructing non-Riemannian Einstein manifolds. In this framework, the torsion of a connection is intrinsically related to the global topology of the manifold, represented by the de Rham cohomology class specified by a set of real parameters. We focus on the product manifold $S^2 \times T^2$, whose third cohomology group is $H^3(S^2 \times T^2; \mathbb{R}) \cong \mathbb{R}^2$. We analyze how the geometry is modeled by the choice of the torsion tensor $T$ within the family $\mathcal{T}_\omega$, defined by the property that each member of this family must have an associated 3-form $T^\flat$ such that it represents the nontrivial cohomology class via Hodge decomposition. Our analysis reveals a dependence on this choice. First, we show that using a torsion tensor, which produces a strictly positive biorthogonal curvature, leads to a non-diagonal Ricci tensor, creating a structural obstacle to any Einstein solution. Conversely, we then show that using the torsion tensor associated with the purelly harmonic 3-form allow us the construction of an explicit non-Riemannian Einstein solution. Our work thus demonstrates that the cohomologically calibrated affine connections allow a family of feasible connections rich enough to allow for several geometries intrinsically justified by the differential topology of the manifold.