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 $m\in M$, 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.