Homogeneous symplectic half-flat 6-manifolds

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.2010 Mathematics Subject Classification. 53C30, 53C15, 53D05.

On the automorphism group of a closed G2-structure

“…In contrast to the last result, it is possible to exhibit non-compact homogeneous examples. Consider for instance a six-dimensional non-compact homogeneous space H/K endowed with an invariant symplectic half-flat SU(3)-structure, namely an SU(3)-structure (ω, ψ) such that dω = 0 and dψ = 0 (see [20] for the classification of such spaces when H is semisimple and for more information on symplectic half-flat structures). If (ω, ψ) is not torsion-free, i.e., if d(Jψ) = 0, then the non-compact homogeneous space (H× S 1 )/K admits an invariant closed non-parallel G 2 -structure defined by the 3-form…”

On the automorphism group of a symplectic half-flat 6-manifold

“…For instance, it is possible to exhibit non-compact examples which are homogeneous under the action of a semisimple Lie group of automorphisms (see e.g. [16]). Moreover, in the next section we shall construct non-compact examples of cohomogeneity one with respect to a semisimple Lie group of automorphisms.…”

“…In mathematical literature, symplectic half-flat structures were first introduced and studied in [6] and then in [8], while explicit examples were exhibited in [5,7,9,16,20]. Most of them consist of simply connected solvable Lie groups endowed with a left-invariant symplectic half-flat structure.…”

“…In [16], we proved the non-existence of compact homogeneous examples and we classified all non-compact cases which are homogeneous under the action of a semisimple Lie group of automorphisms. Here, in Theorem 2.1 we show that the Lie algebra of Aut(M, ω, ψ) is abelian with dimension bounded by min{5, b 1 (M )} whenever M is compact.…”

Special solutions to the Type IIA flow

“…In this last case, the examples are locally homogeneous. Further non-compact homogeneous and cohomogeneity one examples are given in [23,24]. More details can be found in Section 3.…”