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Journal of the London Mathematical Society Advance Access published online on July 10, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdn044
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© 2008 London Mathematical Society

Nearly hypo structures and compact nearly Kähler 6-manifolds with conical singularities

Marisa Fernández

Universidad del País Vasco
Facultad de Ciencia y Tecnología
Departamento de Matemáticas
Apartado 644
48080 Bilbao
Spain

Stefan Ivanov

University of Sofia ‘St. Kl. Ohridski’
Faculty of Mathematics and Informatics
5, James Bourchier Boulevard
1164 Sofia
Bulgaria
ivanovsp@fmi.uni-sofia.bg

Vicente Muñoz

Instituto de Ciencias Matemáticas
CSIC-UAM-UC3M-UCM
Consejo Superior de Investigaciones Científicas
Serrano 113bis
28006 Madrid
Spain
vicente.munoz@imaff.cfmac.csic.es

Luis Ugarte

Departamento de Matemáticas – IUMA
Universidad de Zaragoza
Campus Plaza San Francisco
50009 Zaragoza
Spain
ugarte@unizar.es

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 x R, and a compact nearly Kähler structure with conical singularities on N5 x [0, {pi}] when N5 is compact, thus providing a link between the Calabi–Yau structure on the cone N5 x [0, {pi}] and the nearly Kähler structure on the sin-cone N5 x [0, {pi}]. We define the notion of nearly hypo structure, which leads to a general construction of nearly Kähler structure on N5 x 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 x R given by Bilal and Metzger [Nuclear Phys. B 663 (2003) 343–364]. For N5 = S5 sub S6 and for N5 = S2 x S3 sub S3 x S3, we describe explicitly a Sasaki–Einstein hypo structure as well as the corresponding nearly Kähler structures on N5 x R and N5 x [0, {pi}], and the nearly parallel G2-structures on N5 x R2 and (N5 x [0, {pi}]) x [0, {pi}].

2000 Mathematics Subject Classification 53C15, 53C25, 53C42, 58A15.

Received July 13, 2006; revised April 18, 2008;
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