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

• Autores: María Luisa Fernández Rodríguez , Stefan Ivanov , Vicente Muñoz Velázquez , Luis Ugarte Vilumbrales
• Localización: Journal of the London Mathematical Society, ISSN 0024-6107, Vol. 78, Nº 3, 2008, págs. 580-604
• Idioma: inglés
• DOI: 10.1112/jlms/jdn044
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• Resumen

  • 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 , and a compact nearly Kähler structure with conical singularities on N5 x [0, ] when N5 is compact, thus providing a link between the Calabi�Yau structure on the cone N5 x [0, ] and the nearly Kähler structure on the sin-cone N5 x [0, ]. We define the notion of nearly hypo structure, which leads to a general construction of nearly Kähler structure on N5 x . 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 given by Bilal and Metzger [Nuclear Phys. B 663 (2003) 343�364]. For N5 = S5 S6 and for N5 = S2 x S3 S3 x S3, we describe explicitly a Sasaki�Einstein hypo structure as well as the corresponding nearly Kähler structures on N5 x and N5 x [0, ], and the nearly parallel G2-structures on N5 x 2 and (N5 x [0, ]) x [0, ]