Homogeneous symplectic half-flat 6-manifolds
Fabio Podestà, Alberto Raffero
TL;DR
The authors classify homogeneous SHF 6-manifolds under transitive automorphism groups with semisimple G. They show compact cases must be flat, and they give a complete noncompact semisimple classification: $M\cong \mathrm{SU}(2,1)/\mathrm{T}^2$ (1-parameter SHF family) or $M\cong \mathrm{SO}(4,1)/\mathrm{U}(2)$ (unique up to homothety), with all invariant SHF structures yielding a $J$-Hermitian Ricci tensor. The work employs invariant tensor methods on homogeneous spaces, root-space decompositions, and a detailed analysis of the intrinsic torsion and stability of 3-forms, connecting the models to twistor spaces of hyperbolic 4-manifolds. The results provide explicit SHF data, show consistency of the Ricci tensor with the almost complex structure, and identify the geometric origin of the noncompact examples as twistor spaces $\mathbb{CH}^2$ and $\mathbb{RH}^4$. These findings contribute to the broader understanding of SHF geometry and its links to $G_2$-holonomy flows and related geometric structures in mathematical physics.
Abstract
We consider 6-manifolds endowed with a symplectic half-flat SU(3)-structure and acted on by a transitive Lie group G of automorphisms. We review a classical result allowing to show the non-existence of compact non-flat examples. In the noncompact setting, we classify such manifolds under the assumption that G is semisimple. Moreover, in each case we describe all invariant symplectic half-flat SU(3)-structures up to isomorphism, showing that the Ricci tensor is always Hermitian with respect to the induced almost complex structure. This last condition is characterized in the general case.