We introduce spin-harmonic structures, a class of geometric structures on Riemannian manifolds of low dimension which are defined by a harmonic unitary spinor. Such structures are related to SU(2) (dim = 4, 5), SU(3) (dim = 6) and G 2 (dim = 7) structures; in dimension 8, a spin-harmonic structure is equivalent to a balanced Spin(7) structure. As an application, we obtain examples of compact 8-manifolds endowed with non-integrable Spin(7) structures of balanced type.

We describe the different classes of Spin(7) structures in terms of spinorial equations. We relate them to the spinorial description of G 2 structures in some geometrical situations. Our approach enables us to analyze invariant Spin(7) structures on quasi abelian Lie algebras. IntroductionBerger's list [2] (1955) of possible holonomy groups of simply connected, irreducible and non-symmetric Riemannian manifolds contains the so-called exceptional holonomy groups, G 2 and Spin (7), which occur in dimensions 7 and 8 respectively. Non-complete metrics with exceptional holonomy were given by Bryant in [3], complete metrics were obtained by Bryant and Salamon in [4], but compact examples were not constructed until 1996, when Joyce published [12], [13] and [14].The remaining groups of Berger's list different from SO(n), called special holonomy groups, are U(n), SU(n), Sp(n) and Sp(n) · Sp(1). If the holonomy of a Riemannian manifold is contained in a group G, the manifold admits a G structure, that is, a reduction to G of its frame bundle. Therefore, holonomy is homotopically obstructed by the presence of G structures. Examples of manifolds endowed with G structures for some of the holonomy groups in the Berger list are not only easier to obtain than manifolds with holonomy in G, but also relevant in M-theory, especially if they admit a characteristic connection [10], that is, a metric connection with totally skew-symmetric torsion whose holonomy is contained in G. It is worth mentioning that Ivanov proved in [11] that each manifold with a Spin(7) structure admits a unique characteristic connection. Moreover, Friedrich proved in [9] that Spin (7) is the unique compact simple Lie group G such that all the G structures admit a unique characteristic connection.The Lie group G 2 is compact, simple and simply connected. It consists of the endomorpisms of R 7 which preserve the cross product from the imaginary part of the octonions [22]. Hence, a G 2 structure on a manifold Q determines a 3-form Ψ, a metric and an orientation. In [7], Fernández and Gray classify G 2 structures into 16 different classes in terms of the G 2 irreducible components of ∇Ψ. Related to this, the analysis of the intrinsic torsion in [5] allowed to obtain equations involving dΨ and d( * Ψ) for each of the 16 classes, determined by the G 2 irreducible components of Λ 4 T * Q and Λ 5 T * Q. In particular, one obtains that the holonomy of Q is contained in G 2 if and only if dΨ = 0 and d( * Ψ) = 0. The Lie group Spin (7) is also compact, simple and simply connected. It is the group of endomorphisms of R 8 which preserve the triple cross product from the octonions [22]. Thus, a Spin(7) structure on a manifold M determines 4-form Ω, a metric and an orientation. In [6], Fernández classifies Spin(7) structures into 4 classes in terms of differential equations for dΩ, which are determined by the Spin(7)-irreducible components of Λ 5 T * M . Parallel structures verify dΩ = 0, locally conformally parallel structures satisfy dΩ = θ ∧ Ω for a closed 1-form θ and ...

We prove that any small enough neighborhood of a closed contact submanifold is always tight under a mild assumption on its normal bundle. The non-existence of C 0 -small positive loops of contactomorphisms in general overtwisted manifolds is shown as a corollary.

No abstract

We show that the compact quotient $$\Gamma \backslash {\mathrm G}$$ Γ \ G of a seven-dimensional simply connected Lie group $${\mathrm G}$$ G by a co-compact discrete subgroup $$\Gamma \subset {\mathrm G}$$ Γ ⊂ G does not admit any exact $${\mathrm G}_2$$ G 2 -structure which is induced by a left-invariant one on $${\mathrm G}$$ G .