SOME GENERAL, ALGEBRAIC REMARKS ON TENSOR CLASSIFICATION, THE GROUP O(2,2) AND SECTIONAL CURVATURE IN 4-DIMENSIONAL MANIFOLDS OF NEUTRAL SIGNATURE

This paper presents a general discussion of the geometry of a manifold M of dimension 4 which admits a metric g of neutral signature (+, +, -, -). The tangent space geometry at m is an element of M, the complete pointwise algebraic classification of second order symmetric and skew-symmetric tensors and the algebraic structure of the members of the orthogonal group O(2, 2) are given in detail. The sectional curvature function for (M, g) is also discussed and shown to be an essentially equivalent structure on M to the metric g in all but a few very special cases, and these special cases are briefly introduced. Some brief remarks on the Weyl conformal tensor, Weyl's conformal theorem and holonomy for (M, g) are also given.