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Transactions of the American Mathematical Society

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

 
 

 

Euler-Poincaré characteristic and higher order sectional curvature. I


Authors: Chuan-Chih Hsiung and Kenneth Michael Shiskowski
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
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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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Article copyright: © Copyright 1988 American Mathematical Society