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?
Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
|
6
1
|
|||||||||
|
