TITLE: Einstein Manifolds in dimension five?
QUESTION [8 upvotes]: My teacher of Riemannian Geometry told the class about the problem of finding Einstein Manifolds in 5 dimension. My question is, what is the difficulty of this problem? Possessing a first course in Riemmanian geometry can I understand this?
Thanks

REPLY [4 votes]: Simply connected 5-manifolds are classified by simple invariants, second homology and second Stiefel-Whitney class.  So one can consider the existence problem, i.e. which 5-manifolds admit an Einstein metric.  Many are know to have an Eintein metric.  See the paper by János Kollár "Einstein metrics on 5-dimensional Seifert bundles".