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On a generalized conjecture of Hopf with symmetry
- Part of
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- Journal:
- Compositio Mathematica / Volume 153 / Issue 2 / February 2017
- Published online by Cambridge University Press:
- 01 February 2017, pp. 313-322
- Print publication:
- February 2017
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- Article
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A famous conjecture of Hopf states that
$\mathbb{S}^{2}\times \mathbb{S}^{2}$ does not admit a Riemannian metric with positive sectional curvature. In this article, we prove that no manifold product 
$N\times N$ can carry a metric of positive sectional curvature admitting a certain degree of torus symmetry.
