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

Püttmann, Thomas; Searle, Catherine

doi:10.1090/S0002-9939-01-06039-7

The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank
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by Thomas Püttmann and Catherine Searle PDF

Proc. Amer. Math. Soc. 130 (2002), 163-166 Request permission

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 $k - (\operatorname {rank} G - \operatorname {rank} H)\le 5$. 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 $\ge l$.

References

Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4
Marcel Berger, Riemannian geometry during the second half of the twentieth century, University Lecture Series, vol. 17, American Mathematical Society, Providence, RI, 2000. Reprint of the 1998 original. MR 1729907, DOI 10.1090/ulect/017
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
Robert Geroch, Positive sectional curvatures does not imply positive Gauss-Bonnet integrand, Proc. Amer. Math. Soc. 54 (1976), 267–270. MR 390961, DOI 10.1090/S0002-9939-1976-0390961-8
Karsten Grove and Catherine Searle, Positively curved manifolds with maximal symmetry-rank, J. Pure Appl. Algebra 91 (1994), no. 1-3, 137–142. MR 1255926, DOI 10.1016/0022-4049(94)90138-4
H. Hopf, Über die Curvatura integra geschlossener Hyperflächen, Math. Ann. 95 (1925), 340–367.
J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4
Shoshichi Kobayashi, Transformation groups in differential geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70, Springer-Verlag, New York-Heidelberg, 1972. MR 0355886
Lawrence M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939), 656–660. MR 99
X. Rong, Positively curved manifolds with almost maximal symmetry rank, to appear.
Nolan R. Wallach, Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. of Math. (2) 96 (1972), 277–295. MR 307122, DOI 10.2307/1970789