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Let M 2n be a compact Riemannian manifold of non-positive sectional curvature. It is shown that if M 2n is homeomorphic to a Kähler manifold, then its Euler number satisfies the inequality (−1) n χ(M 2n ) ≥ 0. M 2n of negative sectional curvature must satisfy the inequality (−1) n χ(M 2n ) > 0. This conjecture is true in dimensions 2 and 4 [Ch] and it has been verified in the Kähler case for all n by Gromov [G] and Stern [S] (the work in [S] also uses results of Greene and Wu; see [GW], p.183-215). Gromov's arguments are rather general and establish the following result: "Let M 2n be a compact Riemannian manifold of negative curvature. If M 2n is homotopy equivalent to a compact Kähler manifold then (−1) n χ(M 2n ) > 0"; see [G], Theorem 0.4.A and Example (a), p.265.A companion conjecture asserts that, if the sectional curvature of a Riemannian manifold M 2n is assumed to be only non-positive, then the Euler number must satisfy (−1) n χ(M 2n ) ≥ 0. Again, this second conjecture is known to be true in dimensions two and four [Ch]. The aim of this paper is to establish its validity for all n in the Kähler case, thus complementing the above result of Gromov:Main Theorem. Let M 2n be a compact Riemannian manifold of non-positive curvature. If M 2n is homeomorphic to a Kähler manifold, then the Euler number of M 2n satisfies the inequality (−1) n χ(M 2n ) ≥ 0.Remark. The proof of the Main Theorem works if the two manifolds are assumed to be only homotopy equivalent but, in view of Farrell and Jones [FJ], the man-

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It is a classical theorem of Hilbert that the hyperbolic plane cannot be realized isometrically in ~3. Remark. For n = 2, b) is Efimov's original result, the sectional curvature condition being then superfluous. The complete hypersurfaces of non-negative Ricci curvature and constant mean curvature were shown by Cheng and Yau [1] to be generalized cylinders i.e. products of round spheres with linear spaces. For the case of non-positive Ricci curvature the corresponding result is true, and so classifies all complete hypersurfaces of constant mean curvature (4:0) whose Ricci curvature does not change sign, as generalized cylinders. For n=2 this is due to Klotz and Osserman [3] but their proof uses the