Homogeneous symplectic half-flat 6-manifolds

TL;DR

This paper classifies noncompact homogeneous symplectic half-flat 6-manifolds with semisimple symmetry groups, detailing their invariant structures and Ricci tensor properties, and reviews classical non-existence results for compact cases.

Contribution

It provides a classification of noncompact homogeneous symplectic half-flat SU(3)-structures with semisimple automorphism groups, including explicit descriptions and Ricci tensor analysis.

Findings

Non-existence of compact non-flat examples confirmed.

Complete classification of noncompact homogeneous cases achieved.

Ricci tensor is always Hermitian with respect to the induced almost complex structure.

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.