Homogeneous generalized Einstein manifolds in dimension four

Low-dimensional Homogeneous Einstein Manifolds

A closed Riemannian manifold (Mn, g) is called Einstein if the Ricci tensor of g is a multiple of itself; that is, ric(g) = λ · g. This equation, called the Einstein equation, is a complicated system of second order partial differential equations, and at the present time no general existence results for Einstein metrics are known. However, there are results for many interesting classes of Einst...

Four - Manifolds without Einstein

On Sasaki-einstein Manifolds in Dimension Five

We prove the existence of Sasaki-Einstein metrics on certain simply connected 5-manifolds where until now existence was unknown. All of these manifolds have non-trivial torsion classes. On several of these we show that there are a countable infinity of deformation classes of Sasaki-Einstein structures.

Four-manifolds without Einstein Metrics

It is shown that there are infinitely many compact simply connected smooth 4-manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict Hitchin-Thorpe inequality 2χ > 3|τ |. The examples in question arise as non-minimal complex algebraic surfaces of general type, and the method of proof stems from Seiberg-Witten theory.

Homogeneous Einstein metrics on Stiefel manifolds

A Stiefel manifold VkR n is the set of orthonormal k-frames inR, and it is diffeomorphic to the homogeneous space SO(n)/SO(n−k). We study SO(n)-invariant Einstein metrics on this space. We determine when the standard metric on SO(n)/SO(n−k) is Einstein, and we give an explicit solution to the Einstein equation for the space V2R.