As far as I know, in a simply connected compact manifold, still there exists no well-known obstruction for a manifold with a **quasi-positive curvature** to be a manifold with **positive curvature**. 


But Hopf's conjecture is unsolved, i.e., the conjecture: $S^2\times S^2$ has a positive curvature. 


So I think that the quasi-positive curvature-condition can be weakened by conditions : **non-negative curvature** and non-flatness. 


Here my concrete question is : Is there a simply connected compact simple 
Lie group which does not have a metric with a positive curvature and has a metric with a non-negative curvature ?


[Definitions]


Here we say that a Riemannian manifold $M$ has positive (resp. non-negative) curvature if all sectional curvatures are positive (resp. non-negative) at all points of $M$ And a manifold with quasi-positive curvature is a manifold with a non-negative curvature and a point at which all tangent 2-planes have positive curvature. 



Manifold with a quasi-positive curvature
For high rank Lie group, this is in fact the generalized Hopf conjecture which is still open.


