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- Title
Nearly hypo structures and compact nearly Kahler 6-manifolds with conical singularities.
- Authors
Marisa Fernández; Stefan Ivanov; Vicente Muñoz; Luis Ugarte
- Abstract
We prove that any totally geodesic hypersurface N5 of a 6-dimensional nearly Kähler manifold M6 is a Sasaki–Einstein manifold, and so it has a hypo structure in the sense of Conti and Salamon [Trans. Amer. Math. Soc. 359 (2007) 5319–5343]. We show that any Sasaki–Einstein 5-manifold defines a nearly Kähler structure on the sin-cone N5 × ℝ, and a compact nearly Kähler structure with conical singularities on N5 × [0, π] when N5 is compact, thus providing a link between the Calabi–Yau structure on the cone N5 × [0, π] and the nearly Kähler structure on the sin-cone N5 × [0, π]. We define the notion of nearly hypo structure, which leads to a general construction of nearly Kähler structure on N5 × ℝ. We characterize double hypo structure as the intersection of hypo and nearly hypo structures and classify double hypo structures on 5-dimensional Lie algebras with non-zero first Betti number. An extension of the concept of nearly Kähler structure is introduced, which we refer to as nearly half-flat SU(3)-structure, and which leads us to generalize the construction of nearly parallel G2-structures on M6 × ℝ given by Bilal and Metzger [Nuclear Phys. B 663 (2003) 343–364]. For N5 = S5 ⊂ S6 and for N5 = S2 × S3 ⊂ S3 × S3, we describe explicitly a Sasaki–Einstein hypo structure as well as the corresponding nearly Kähler structures on N5 × ℝ and N5 × [0, π], and the nearly parallel G2-structures on N5 × ℝ2 and (N5 × [0, π]) × [0, π].
- Subjects
MANIFOLDS (Mathematics); HYPERSURFACES; EINSTEIN manifolds; MATHEMATICAL singularities; MATHEMATICAL analysis; HYPERSPACE
- Publication
Journal of the London Mathematical Society, 2008, Vol 78, Issue 3, p580
- ISSN
0024-6107
- Publication type
Article