2025/08/04 by Pigazzini, Alexander, Toda, Magdalena · 1 citation
#53C05 #53C15 #53C25 #58A14 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2508.02353
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 S2 × T2, whose third cohomology group is H3(S2 × T2; ℝ) ≅ ℝ2. We analyze how the geometry is modeled by the choice of the torsion tensor T within the family Tω, 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.