Special Symplectic Six-Manifolds

Abstract

We classify nilmanifolds with an invariant symplectic half-flat structure. We study the transverse or quotient geometry of six-manifolds with an SU(3)-structure preserved by a Killing vector field, giving characterizations in the symplectic half-flat and integrable case.

Contact 5-Manifolds With Su(2)-Structure

“…We must notice that this result, which is also used in [10], implies that the holonomy of the resulting G 2 -metric on M × I is contained in SU (3), because it is actually a product metric. This fact justifies our study of half-flat structures on non-trivial circle bundles (see Remark 6.2 below).…”

The Ricci tensor of SU(3)-manifolds

“…SGCY manifolds are taken into consideration also in [8], [15] and [25]. Such a structure can be characterized by…”

GENERALIZED G2-MANIFOLDS AND SU(3)-STRUCTURES

“…In this context, Witt [17] introduced a new type of structures on a 7-dimensional manifold M in terms of a differential form of mixed degree, thus generalizing the classical notion of G 2 -structure determined by a stable and positive 3-form. Instead of studying geometry on the tangent bundle T M of the manifold, one considers the bundle T M ⊕ T * M endowed with a natural orientation and an inner product of signature (7,7), where T * M denotes the cotangent bundle of M . In this way, if M is spin, then the differential form of mixed type can be viewed as a G 2 × G 2 -invariant spinor ρ for the bundle and it is called the structure form.…”

“…The existence of weakly integrable generalized G 2 -structures with respect to a closed 3-form H on a compact manifold was posed as an open problem in [17]. We construct such structures on a family of compact manifolds and we relate them with SU (3)-structures in dimension 7, where SU (3) is identified with the subgroup SU (3) × {1} of SO (7).…”

“…Indeed, starting with a half-flat structure on N , if certain evolution equations are satisfied, then there exists a Riemannian metric with holonomy contained in G 2 on the product of the manifold N with some open interval. Examples of compact manifolds with symplectic half-flat structures have been given in [7], where invariant symplectic half-flat structures on nilmanifolds are classified. Other examples are considered in [8] where Lagrangian submanifolds are studied instead.…”