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Kleiner-Hsiang (JDG 1989) proved such a manifold is homeomorphic to $S^4$ or $CP^{2}$. an interesting corrollary is that $S^2 \times S^2$ does not admit positively curved metric with countinuous symmetry. They asked in their paper if it is diffeomorphic. I don't know much on this area, only noticed one subsequent work Searle-Yang. Seems that there are preprints on arXiv attempting to do the problem in full generality. Not sure if it was fully settled. Anyone knows the precise status of this problem?

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One preprint seems to have appeared in publication: – Ian Agol Feb 4 '12 at 17:24
Thanks for pointing it out. I am just not sure if that paper is well acknowledged. – Bo_Y Feb 9 '12 at 3:11
Perhaps it is worth reading the review on MR that R. Fintushel wrote about the paper quoted above: – Renato G. Bettiol Nov 15 '13 at 2:50

The status of the problem is that it is solved: the classification up to (equivariant) diffeomorphism was obtained very recently by K. Grove and B. Wilking, see Thm A here. The proof uses the solution of the Poincare' Conjecture and some fancy machinery of Alexandrov geometry.

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