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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
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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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  • [A] C.B. Allendoerfer, The Euler number of a Riemannian manifold, Amer. J. Math. 62 (1940), 243-248. MR 2:20e
  • [AW] C.B. Allendoerfer, A. Weil, The Gauss-Bonnet theorem for Riemannian polyhedra, Trans. Amer. Math. Soc. 53 (1943), 101-129. MR 4:169e
  • [Be] M. Berger, Riemannian geometry during the second half of the twentieth century, University Lecture Series, 17, American Mathematical Society, Providence, RI, 1998. MR 2000h:53002
  • [Br] G.E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London 1972. MR 54:1265
  • [C1] S.S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. (2) 45 (1944), 747-752. MR 6:106a
  • [C2] S.S. Chern, On curvature and characteristic classes of a Riemann manifold, Abh. Math. Sem. Univ. Hamburg 20 (1955), 117-126. MR 17:783e
  • [F] W. Fenchel, On total curvatures of Riemannian manifolds: I, J. London Math. Soc. 15 (1940), 15-22. MR 2:20f
  • [G] R. Geroch, Positive sectional curvature does not imply positive Gauss-Bonnet integrand, Proc. Amer. Math. Soc. 54 (1976), 267-270. MR 52:11784
  • [GS] K. Grove, C. Searle, Positively curved manifolds with maximal symmetry-rank, J. Pure Appl. Algebra 91 (1994), 137-142. MR 95i:53040
  • [H] H. Hopf, Über die Curvatura integra geschlossener Hyperflächen, Math. Ann. 95 (1925), 340-367.
  • [HS] H. Hopf, H. Samelson, Ein Satz über die Wirkungsräume geschlossener Liescher Gruppen, Comment. Math. Helv. 13 (1941), 240-251. MR 4:3b
  • [K] S. Kobayashi, Transformation groups in differential geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70, Springer, Berlin-Heidelberg-New York 1972. MR 50:8360
  • [PV] F. Podestà, L. Verdiani, Totally geodesic orbits of isometries, Ann. Global Anal. Geom. 16 (1998), 413-418. MR 99j:53071
  • [R] X. Rong, Positively curved manifolds with almost maximal symmetry rank, to appear.
  • [W] N.R. Wallach, Compact homogeneous Riemannian manifolds with strictly positive sectional curvature, Ann. of Math. 96 (1972), 277-295. MR 46:6243

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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

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