Abstract
We investigate a new 8-dimensional Riemannian geometry defined by a generic closed and coclosed 3-form with stabiliser PSU(3), and which arises as a critical point of Hitchin's variational principle. We give a Riemannian characterisation of this structure in terms of invariant spinor-valued 1-forms, which are harmonic with respect to the twisted Dirac operator D on Delta circle times Lambda(1). ...
Abstract
We investigate a new 8-dimensional Riemannian geometry defined by a generic closed and coclosed 3-form with stabiliser PSU(3), and which arises as a critical point of Hitchin's variational principle. We give a Riemannian characterisation of this structure in terms of invariant spinor-valued 1-forms, which are harmonic with respect to the twisted Dirac operator D on Delta circle times Lambda(1). We establish various obstructions to the existence of topological reductions to PSU(3). For compact manifolds, we also give sufficient conditions for topological PSU(3)-structures that can be lifted to topological SU (3)-structures. We also construct the first known compact example of an integrable non-symmetric PSU(3)-structure. In the same vein, we give a new Riemannian characterisation for topological quaternionic Kahler structures which are defined by an Sp(l) . Sp(2)-invariant self-dual 4-form. Again, we show that this form is closed if and only if the corresponding spinor-valued 1-form is harmonic for D and that these equivalent conditions produce constraints on the Ricci tensor. (C) 2008 Elsevier Inc. All fights reserved.
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