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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Euler-Poincaré characteristic and higher order sectional curvature. I
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by Chuan-Chih Hsiung and Kenneth Michael Shiskowski PDF
Trans. Amer. Math. Soc. 305 (1988), 113-128 Request permission

Abstract:

The following long-standing conjecture of H. Hopf is well known. Let $M$ be a compact orientable Riemannian manifold of even dimension $n \geqslant 2$. If $M$ has nonnegative sectional curvature, then the Euler-Poincaré characteristic $\chi (M)$ is nonnegative. If $M$ has nonpositive sectional curvature, then $\chi (M)$ is nonnegative or nonpositive according as $n \equiv 0$ or $2\bmod 4$. This conjecture for $n = 4$ was proved first by J. W. Milnor and then by S. S. Chern by a different method. The main object of this paper is to prove this conjecture for a general $n$ under an extra condition on higher order sectional curvature, which holds automatically for $n = 4$. Similar results are obtained for Kähler manifolds by using holomorphic sectional curvature, and F. Schur’s theorem about the constancy of sectional curvature on a Riemannian manifold is extended.
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 305 (1988), 113-128
  • MSC: Primary 53C20; Secondary 53C55, 57R20
  • DOI: https://doi.org/10.1090/S0002-9947-1988-0920149-3
  • MathSciNet review: 920149