On a generalized conjecture of Hopf with symmetry
  A famous conjecture of Hopf is that the product of the two-dimensional sphere with itself does not admit a Riemannian metric with positive sectional curvature. More generally, one may conjecture that this holds for any nontrivial product. We provide evidence for this generalized conjecture in the presence of symmetry. 
Manuel Amann and Lee Kennard
math.DG