Sasakian Geometry

This book is an extensive monograph on Sasakian manifolds , focusing on the intricate relationship between K"ahler and Sasakian geometries. The subject is introduced by discussion of several background topics, including the theory of Riemannian foliations, compact complex and K"ahler orbifolds, and the existence and and obstruction theory of K"ahler-Einstein metrics on complex compact orbifolds. There is then a discussion of contact and almost contact structures in the Riemannian setting, in which compact quasi-regular Sasakian manifolds emerge as algebraic objects. There is an extensive discussion of the symmetries of Sasakian manifolds, leading to a study of Sasakian structures on links of isolated hypersurface singularities. This is followed by an in-depth study of compact sasakian manifolds in dimensions three and five. The final section of the book deals with the existence of Sasaki-Einstein metrics. 3-Sasakian manifolds and the role of sasakian-Einstein geometry in String Theory are discussed separately.