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

DOI:
https://doi.org/10.1090/S0002-9939-01-06039-7

Published electronically:
June 8, 2001

MathSciNet review:
1855634

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 with principal isotropy group and cohomogeneity such that . Moreover, we prove that the Euler characteristic of a compact Riemannian manifold or with positive sectional curvature is positive if admits an effective isometric action of a torus , i.e., if the symmetry rank of is .

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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:
https://doi.org/10.1090/S0002-9939-01-06039-7

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