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Nearly hypo structures and compact nearly Kähler 6‐manifolds with conical singularities
Author(s) -
Fernández Marisa,
Ivanov Stefan,
Muñoz Vicente,
Ugarte Luis
Publication year - 2008
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdn044
Subject(s) - hypersurface , manifold (fluid mechanics) , holonomy , mathematics , pure mathematics , einstein , gravitational singularity , cone (formal languages) , physics , zero (linguistics) , conical surface , mathematical physics , geodesic , mathematical analysis , geometry , mechanical engineering , linguistics , philosophy , algorithm , engineering
We prove that any totally geodesic hypersurface N 5 of a 6‐dimensional nearly Kähler manifold M 6 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 N 5 × ℝ , and a compact nearly Kähler structure with conical singularities on N 5 × [0, π] when N 5 is compact, thus providing a link between the Calabi–Yau structure on the cone N 5 × [0, π] and the nearly Kähler structure on the sin‐cone N 5 × [0, π]. We define the notion of nearly hypo structure, which leads to a general construction of nearly Kähler structure on N 5 × 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 G 2 ‐structures on M 6 × ℝ given by Bilal and Metzger [ Nuclear Phys. B 663 (2003) 343–364]. For N 5 = S 5 ⊂ S 6 and for N 5 = S 2 × S 3 ⊂ S 3 × S 3 , we describe explicitly a Sasaki–Einstein hypo structure as well as the corresponding nearly Kähler structures on N 5 × ℝ and N 5 × [0, π], and the nearly parallel G 2 ‐structures on N 5 × ℝ 2 and ( N 5 × [0, π]) × [0, π].