Neutral Einstein metrics in four dimensions

Journal Article · Fri Nov 01 00:00:00 EST 1991 · Journal of Mathematical Physics (New York); (United States)

DOI:https://doi.org/10.1063/1.529048· OSTI ID:5019656

Law, P R ^[1]

---

1. Department of Mathematics, Tufts University, Medford, Massachusetts (USA)

In Matsushita (J. Math. Phys. {bold 22}, 979--982 (1981), {ital ibid}. {bold 24}, 36--40 (1983)), for curvature endomorphisms for the pseudo-Euclidean space {ital R}{sup 2,2}, 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 {bold 35}, 17--26 (1963) and {ital Shiing}-{ital Shen} {ital Chern}: {ital Selected} {ital 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.

• OSTI ID: 5019656
• Journal Information: Journal of Mathematical Physics (New York); (United States), Vol. 32:11; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
METRICS
FOUR-DIMENSIONAL CALCULATIONS
DIFFERENTIAL GEOMETRY
GENERAL RELATIVITY THEORY
GEOMETRY
GRAVITATION
RIEMANN SPACE
SPACE-TIME
TENSORS
FIELD THEORIES
MATHEMATICAL SPACE
MATHEMATICS
SPACE
657003* - Theoretical & Mathematical Physics- Relativity & Gravitation