Yau asked in 1982 if there is *any* compact simply connected manifold with nonnegative curvature for which one can prove that it does not admit a metric of positive curvature. This question opens his list of unsolved problems in geometry (see [*"Seminar on Differential Geometry"*][1], P.670.)  


Let me quote from [*"A Panoramic View of Riemannian Geometry*"][2] by Berger (Springer 2003, p. 579):

>It is not surprising that many people tried to address Yau’s remark, starting
with the Hopf conjecture on $S^2 × S^2$, by trying to deform such a metric
with $K ≥ 0$ into one with $K >0$. This means considering some one parameter
family $g(t)$ of metrics and computing the various derivatives at $t = 0$ of the
sectional curvature. Technically it is very easy to compute such a derivative
for a given tangent plane, but what is difficult is to find a variation for which
all the derivatives would be positive. Today this approach still does not work...

The earlier short [survey][3] by Bourguignon contains a discussion of some of the reasons why seemingly natural approaches fail.


  [1]: http://books.google.co.uk/books?id=Hm9fLdZGQY8C&pg=PA455&lpg=PA455&dq=Yau+Problem+section,+Seminar+on+Differential+Geometry&source=bl&ots=i4hrlW2-AN&sig=35fGdjnx9ZxFI2lfswP5ssDJZE0&hl=en&ei=1bv2TNmHOsmeOpCh5M8I&sa=X&oi=book_result&ct=result&resnum=4&ved=0CCwQ6AEwAw#v=onepage&q=Yau%2520Problem%2520section%252C%2520Seminar%2520on%2520Differential%2520Geometry&f=false
  [2]: http://books.google.co.uk/books?id=d_SsagQckaQC&printsec=frontcover&dq=berger+panoramic+view+of+riemannian+geometry&source=bl&ots=6_yIGcNtK0&sig=KB9VAEfFo4wHN-dgsMwrhobzVFI&hl=en&ei=H7r2TP_JCYOBOqWhxaAI&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBYQ6AEwAA#v=onepage&q&f=false
  [3]: http://books.google.co.uk/books?hl=en&lr=&id=stKz4KYmfi8C&oi=fnd&pg=PA33&ots=Ub8mjq9KJ6&sig=YDhJkWJvufMdPBxrfwXTvB6drwU#v=onepage&q&f=false