Generalized Killing spinors in dimension 5

Abstract

Abstract. 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).

Calabi-Yau cones from contact reduction

“…In the real analytic category, the converse also holds (see [9] for the five-dimensional case, and Proposition 2 for the general case).…”

“…Generalized Killing spinors arise naturally, by restriction, on oriented hypersurfaces in manifolds with a parallel spinor; in this setting, the tensor Q corresponds to the Weingarten tensor. There are also partial results in the converse direction (see [22,2,9]). In particular, consider the case of a hypersurface M inside a manifold with holonomy SU(n + 1).…”

“…However, the fact that (i) implies (ii) does not require this hypothesis. A five-dimensional version of Proposition 2 is proved in [9], where the SU(2)-structures defined by a generalized Killing spinor were introduced under the name of hypo structures. In that paper it was also proved, by considering the intrinsic torsion, that (ii) implies (i) without assuming real analyticity.…”

“…which determines X s because the ω s are symplectic forms, to ensure that (9) holds. With this definition, X s vanishes on M × {0}.…”

Invariant forms, associated bundles and Calabi–Yau metrics

“…We shall use our list of invariant forms to determine other Calabi-Yau structures (not necessarily conical) defined on a neighbourhood of S 1 in T CP 2 whose Kähler form lies in the family (39). The starting observation is the following: if M is a Calabi-Yau manifold of dimension 8, each oriented hypersurface inherits an SU(3)-structure, defined by differential forms F , Ω and α which at each point satisfy F = e 12 + e 34 + e 56 , Ω = (e 1 + ie 2 ) ∧ (e 3 + ie 4 ) ∧ (e 5 + ie 6 ), α = e 7 ; moreover, F and α∧Ω are closed [10]. In the real analytic category, the converse also holds.…”