Contact Structures of Sasaki Type and their Associated Moduli

This article is based on a talk at the RIEMain in Contact conference in Cagliari, Italy in honor of the 78th birthday of David Blair one of the founders of modern Riemannian contact geometry. The present article is a survey of a special type of Riemannian contact structure known as Sasakian geometry. An ultimate goal of this survey is to understand the moduli of classes of Sasakian structures as well as the moduli of extremal and constant scalar curvature Sasaki metrics, and in particular the moduli of Sasaki-Einstein metrics.Comment: 48 page

Sasakian Geometry, Holonomy, and Supersymmetry

In this expository article we discuss the relations between Sasakian geometry, reduced holonomy and supersymmetry. It is well known that the Riemannian manifolds other than the round spheres that admit real Killing spinors are precisely Sasaki-Einstein manifolds, 7-manifolds with a nearly parallel G2 structure, and nearly Kaehler 6-manifolds. We then discuss the relations between the latter two and Sasaki-Einstein geometry.Comment: 40 pages, some minor corrections made, to appear in the Handbook of pseudo-Riemannian Geometry and Supersymmetr

Sasakian Geometry, Hypersurface Singularities, and Einstein Metrics

We review our study of Sasakian geometry as an agent for proving the existence of Einstein metrics on odd dimensional manifolds. Particular emphasis is given to the Sasakian structures occuring on links of isolated hypersurface singularities.Comment: This is an expository article that grew out of notes for the three lectures the second author presented during the XXIV-th Winter School of {\it Geometry and Physics} in Srni, Czech Republic, in January of 2004., 30 pages. Some new examples and references were adde