Construction of metrics on Sasaki-Einstein manifolds

Sasaki-Einstein manifolds are Einstein manifolds occurring in odd dimensions, which have positive curvature and Calabi-Yau metric cones. We start with an introduction of Sasakian manifolds, following the historical development of the subject. For this part, we define almost contact and contact structures and normality. Then, we define Killing spinors and examine their relationship with Sasaki-Einstein manifolds. Afterwards we study two different explicit nonsingular metric constructions of Sasaki-Einstein manifolds, in the form of principle U(1) bundles over Kähler-Einstein base manifolds. We see that even though the base manifolds are singular, we can obtain a nonsingular total space. In the second construction, base manifolds are themselves S^2 bundles over a Kähler-Einstein manifold, whereas in the first one they are S^2 bundles over a product of Kähler-Einstein manifolds. For the first one, we give a detailed analysis in dimension 7 and then a generalization to arbitrary odd dimensions greater than or equal to 7. The second construction applies to all odd dimensions.