Construction of metrics on Sasaki-Einstein manifolds

Abstract

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 de ne almost contact and contact structures and normality. Then, we de ne Killing spinors and examine their relationship with Sasaki-Einstein manifolds. Afterwards we study two di erent explicit nonsingular metric constructions of Sasaki- Einstein manifolds, in the form of principle U(1) bundles over Kahler-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 S2 bundles over a Kahler-Einstein manifold, whereas in the rst one they are S2 bundles over a product of Kahler-Einstein manifolds. For the rst one, we give a detailed analysis in dimension 7 and then a generalization to arbitrary odd dimensions 7. The second construction applies to all odd dimensions.