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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
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