Abstract: We describe a method for constructing Einstein metrics of positive scalar curvature on contact manifolds with metrics naturally adapted to the contact structure. Such spaces are called Sasakian and they are odd-dimensional analogues of Kaehler manifolds. In particular, they have one dimensional foliation (contact or characteristic foliation) whose transverse geometry is Kaehlerian. In the case the adapted metric g is also Einstein, the transverse space is Fano and the transverse metric is Kaehler-Einstein.

It follows that the key to obtaining new Einstein metrics on compact smooth contact manifolds with Sasakian metrics is to establish existence of Kaehler-Einstein metrics on suitable compact Fano orbifolds. This is an orbifold version of a classical Monge-Ampere problem and the well-known continuity method can be extended to work in surprisingly many cases.

Although the method is very general and can be used to prove existence of families of Einstein metrics on many contact manifolds we will demonstrate its power restricting all attention to spheres, more precisely odd-dimensional homotopy spheres. Here the key is the classical differential topology and Sasakian geometry of the famous Brieskorn-Pham links. Working with Brieskorn-Pham orbifolds we are able to show that standard spheres admit large families of Einstein metrics. Furthermore, all homotopy spheres in dimension 7 and 11 admit Einstein metric. Conjecturely all spheres which bound parallelizable manifolds admit Einstein metrics.