NEUTRAL EINSTEIN-METRICS IN 4 DIMENSIONS

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.