ar X iv : m at h / 05 08 37 5 v 2 [ m at h . D G ] 2 9 Ju n 20 06 Generalized Killing Spinors in Dimension 5

We study the intrinsic geometry of hypersurfaces in Calabi-Yau manifolds of real dimension 6 and, more generally, SU(2)-structures on 5-manifolds defined by a generalized Killing spinor. We prove that in the real analytic case, such a 5-manifold can be isometrically embedded as a hypersurface in a Calabi-Yau manifold in a natural way. We classify nilmanifolds carrying invariant structures of this type, and present examples of the associated metrics with holonomy SU(3). MSC classification: 53C25; 14J32, 53C29, 53C42, 58A15 Let N be a spin manifold, and let ΣN be the complex spinor bundle, which splits as Σ+N ⊕ Σ−N in even dimension. It is well known that any oriented hypersurface ι : M → N is also spin, and we have ΣM = ι∗ΣN or ΣM = ιΣ+N according to whether the dimension of N is odd or even. Thus, a spinor ψN on N (which we assume to lie in Σ+N if the dimension is even) induces a spinor ψ = ιψN on M . If ψN is parallel with respect to the Levi-Civita connection of N , then ∇Xψ = 1 2 A(X) · ψ (1) where ∇ is the covariant derivative with respect to the Levi-Civita connection ofM , the dot represents Clifford multiplication and A is a section of the bundle of symmetric endomorphisms of TM ; in fact, A is the Weingarten tensor. On a Riemannian spin manifold, spinors ψ satisfying (1) for some symmetric A are called generalized Killing spinors [4]. Generalized Killing spinors with tr(A) constant arise in the study of the Dirac operator, and are called T -Killing spinors [16]. For a consistent terminology, we define Killing spinors by the condition A = λ Id, where λ is required to be a real constant. If N is the cone on M , i.e. the warped product M ×r R, then ιψ is a Killing spinor. Any generalized Killing spinor ψ is parallel with respect to a suitable connection; consequently, ψ defines a G-structure consisting of those frames u such that ψ = [u, ψ0] for some fixed ψ0 in Σn, where G is the stabilizer of ψ0. The intrinsic torsion of this G-structure can be identified with A. It is easy to prove that the G-structures defined by a generalized Killing spinor