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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank


Authors: Thomas Püttmann and Catherine Searle
Journal: Proc. Amer. Math. Soc. 130 (2002), 163-166
MSC (2000): Primary 53C20
Published electronically: June 8, 2001
MathSciNet review: 1855634
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Abstract:

We prove that the Euler characteristic of an even-dimensional compact manifold with positive (nonnegative) sectional curvature is positive (nonnegative) provided that the manifold admits an isometric action of a compact Lie group $G$ with principal isotropy group $H$ and cohomogeneity $k$ such that $k - (\rank G - \rank H)\le 5$. Moreover, we prove that the Euler characteristic of a compact Riemannian manifold $M^{4l+4}$ or $M^{4l+2}$ with positive sectional curvature is positive if $M$ admits an effective isometric action of a torus $T^l$, i.e., if the symmetry rank of $M$ is $\ge l$.


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Additional Information

Thomas Püttmann
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Email: puttmann@math.ruhr-uni-bochum.de

Catherine Searle
Affiliation: Instituto de Matematicas, Unidad Cuernavaca-UNAM, Apartado Postal 273-3, Admon. 3, Cuernavaca, Morelos, 62251, Mexico
Email: csearle@matcuer.unam.mx

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06039-7
PII: S 0002-9939(01)06039-7
Received by editor(s): May 16, 2000
Published electronically: June 8, 2001
Additional Notes: This paper has its roots in the nonnegative curvature seminar at the University of Pennsylvania in the academic year 1999–2000. The first named author would like to thank the University of Pennsylvania for their hospitality and A. Rigas, B. Wilking, and W. Ziller for advice and many interesting discussions during this time. The second named author was supported in part by a grant from CONACYT project number 28491-E
Communicated by: Wolfgang Ziller
Article copyright: © Copyright 2001 American Mathematical Society



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