The Euclidean-space conjecture for negatively curved homogeneous Einstein manifolds

A homogeneous manifold is a manifold equipped with a transitive group action by isometries, and an Einstein manifold has Ricci tensor proportional to its metric. Its scalar curvature is negative when that proportionality constant is negative. Euclidean-space conjecture. Any connected homogeneous Einstein manifold of negative scalar curvature is diffeomorphic to a Euclidean space. This is described as one of the most important open problems on Einstein homogeneous manifolds. The source also notes that it is equivalent to the analogous statement for expanding algebraic solitons.