3-Sasakian manifolds. (English) Zbl 1008.53047

Boyer, Charles; Galicki, Krzysztof

LeBrun, Claude (ed.) et al., Surveys in differential geometry: essays on Einstein manifolds. Lectures on geometry and topology, sponsored by Lehigh University’s Journal of Differential Geometry. Cambridge, MA: International Press. Surv. Differ. Geom., Suppl. J. Differ. Geom. 6, 123-184 (1999).

This is a review chapter on 3-Sasakian manifolds. First, there are presented definitions, notions and fundamental properties of Sasakian, Sasakian-Einstein and 3-Sasakian manifolds. Then the geometry of the foliations on these manifolds are discussed and a classification of homogeneous structures is given. There are enclosed many important results on Betti numbers of Sasakian and 3-Sasakian manifolds, Killing spinors, \(G_2\)-structures, 3-Sasakian quotient construction and toric 3-Sasakian manifolds. The last part of the paper contains open problems, questions and conjectures followed by an appendix on fundamentals of orbifolds. The reader will find a brief history of 3-Sasakian structures and 134 items as references.
For the entire collection see [Zbl 0961.00021].

Reviewer: Constantin Calin (Iasi)

MSC:

53C25

Special Riemannian manifolds (Einstein, Sasakian, etc.)

Keywords:

3-Sasakian manifolds; Betti numbers; Killing spinors; 3-Sasakian quotient construction