Pure Type-One Generalized Hypercomplex Structures in Dimension Four

Abstract

We classify untwisted generalized hypercomplex structures on four-manifolds whose entire sphere has type one. The three Poisson tensors determine a canonical line field, and the local data are Gauduchon--Tod coframes. We determine local equivalence under diffeomorphisms, common closed real $B$-fields and constant rotations of the triple. On a closed manifold the canonical foliation is automatically Riemannian: a nonzero period obstructs the exact twist required by the non-Riemannian models. The compact structures form four families, distinguished by $b_1=1,2,3,4$: Berger quotients, flat axis quotients, nontrivial circle bundles over flat three-tori, and constant tori. We determine their parameters, equivalences and finite ordered quotients, including nonorientable canonical lines. The underlying manifolds are parallelizable, satisfy $b_2=2b_1-2$, and are symplectic exactly when the intrinsic Gauduchon--Tod invariant vanishes. A twisted Hopf example shows why the untwisted hypothesis is essential. The classification is intrinsic and assumes no initial circle action. These results address the four-dimensional untwisted case of the intermediate-type question of Fino and Grantcharov.