Is there any progress to the famous conjecture of Hopf? There is no metric of positive sectional curvature on $\mathbb{S}^2\times\mathbb{S}^2$. Thanks.
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5$\begingroup$ no $~~~~~~~~~~~~$ $\endgroup$– Anton PetruninCommented Jul 5, 2018 at 10:57
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1$\begingroup$ Andrey Rekalo has an excellent answer here: mathoverflow.net/questions/47942/…. Also, I'd like to emphasize the deformation approach to the Hopf conjecture. Amazingly enough, this question is still completely open: Does there exist a smooth approximation of the standard metric on $\mathbb{S}^2 \times \mathbb{S}^2$ by positively curved metrics? If you want to show there is no positively curved metric on $\mathbb{S}^2\times \mathbb{S}^2$, you might first want to show that the answer to this question is no. $\endgroup$– Alec PayneCommented Jul 5, 2018 at 21:46
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