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

Authors:	Thomas Pü ttmann Catherine Searle

Institution:	Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany ; Instituto de Matematicas, Unidad Cuernavaca-UNAM, Apartado Postal 273-3, Admon. 3, Cuernavaca, Morelos, 62251, Mexico

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 with principal isotropy group and cohomogeneity 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 admits an effective isometric action of a torus , i.e., if the symmetry rank of is $\ge l$ .

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