Hitchin–Thorpe-Type Inequalities for Pseudo-Riemannian 4-Manifolds of Metric Signature (++−−)

Abstract

It is well known that the Euler characteristic χ and the Hirzebruch index τ of a compact Einstein 4-manifold satisfy the Hitchin–Thorpe inequality χ ≥ \(\tfrac{3}{2}\); |τ|, and that these topological invariants of a compact pseudo-Riemannian Einstein 4-manifold of metric signature (++−−), with certain curvature restrictions, also satisfy a similar inequality χ ≥ \(\tfrac{3}{2}\); |τ|, which is called the Hitchin–Thorpe-type inequality. This shows that the Euler characteristic χ is nonpositive for the indefinite case. In this paper, it is shown that for a compact pseudo-Riemannian 4-manifold of signature (++−−) the Hitchin–Thorpe-type inequality holds under a weaker condition, called the diagonal Einstein condition, than the Einstein condition. Our analysis is based on the fact that the existence of a metric of signature (++−−) on a 4-manifold with the structure group SO o (2,2) is equivalent to the existence of a pair of an almost complex structure and an opposite almost complex structure on the 4-manifold, which is also equivalent to the reduction of the structure group to the maximal compact subgroup SO(2) × SO(2) of SO o (2,2).