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On a generalized conjecture of Hopf with symmetry

Part of: Global differential geometry Topological manifolds

Published online by Cambridge University Press:  01 February 2017

Manuel Amann  and
Lee Kennard
Manuel Amann
Affiliation:
Fakultät für Mathematik, Institut für Algebra und Geometrie, Karlsruher Institut für Technologie, Englerstraße 2, Karlsruhe, 76131 Karlsruhe, Germany email manuel.amann@kit.edu
Lee Kennard
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, OK 73019-3103, USA email kennard@math.ou.edu
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Abstract

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.

Keywords

positive curvaturetorus symmetryEuler characteristicHopf conjecture

MSC classification

Primary: 53C20: Global Riemannian geometry, including pinching
Secondary: 57N65: Algebraic topology of manifolds

Information

Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 2 , February 2017 , pp. 313 - 322
Copyright
© The Authors 2017 

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