On Euler characteristics of compact Einstein 4-manifolds of metric signature (++−−)

Abstract

In the present paper, the Euler characteristic is studied for a compact oriented Einstein 4-manifold of signature (++−−). For such a manifold, there are three types of normal forms of the curvature tensor. It is shown that for each type the Euler characteristic is nonnegative and even. Several new inequalities are obtained concerning the Euler characteristic and the volume of the manifold. The second Betti number is even, and if it is zero, then the first Betti number also vanishes. The arguments developed here are based upon the famous work of Chern about the Gauss–Bonnet formula for a pseudo-Riemannian manifold.