Euler-Poincaré characteristic and higher order sectional curvature. I

Abstract

The following long-standing conjecture of H. Hopf is well known. Let M M be a compact orientable Riemannian manifold of even dimension n ⩾ 2 n \geqslant 2 . If M M has nonnegative sectional curvature, then the Euler-Poincaré characteristic χ ( M ) \chi (M) is nonnegative. If M M has nonpositive sectional curvature, then χ ( M ) \chi (M) is nonnegative or nonpositive according as n ≡ 0 n \equiv 0 or 2 mod 4 2\bmod 4 . This conjecture for n = 4 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 n under an extra condition on higher order sectional curvature, which holds automatically for n = 4 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.