Neutral Einstein metrics in four dimensions

Abstract

In Matsushita [J. Math. Phys. 22, 979–982 (1981), ibid. 24, 36–40 (1983)], for curvature endomorphisms for the pseudo-Euclidean space R2,2, an analog of the Petrov classification as a basis for applications to neutral Einstein metrics on compact, orientable, four-dimensional manifolds is provided. This paper points out flaws in Matsushita’s classification and, moreover, that an error in Chern’s [‘‘Pseudo-Riemannian geometry and the Gauss–Bonnet formula,’’ Acad. Brasileira Ciencias 35, 17–26 (1963) and Shiing-Shen Chern: Selected Papers (Springer-Verlag, New York, 1978)] Gauss–Bonnet formula for pseudo-Riemannian geometry was incorporated in Matsushita’s subsequent analysis. A self-contained account of the subject of the title is presented to correct these errors, including a discussion of the validity of an appropriate analog of the Thorpe–Hitchin inequality of the Riemannian case. When the inequality obtains in the neutral case, the Euler characteristic is nonpositive, in contradistinction to Matsushita’s deductions.