Some general, algebraic remarks on tensor classification, the group $O(2,2)$ and sectional curvature in 4-dimensional manifolds of neutral signature

Graham Hall¹

¹ Institute of Mathematics University of Aberdeen Aberdeen AB24 3UE, Scotland, UK

Colloquium Mathematicum, Tome 150 (2017) no. 1, pp. 63-86.

Résumé

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.

Détail

DOI : 10.4064/cm7140s-3-2017

Keywords: paper presents general discussion geometry manifold dimension which admits metric neutral signature tangent space geometry complete pointwise algebraic classification second order symmetric skew symmetric tensors algebraic structure members orthogonal group given detail sectional curvature function discussed shown essentially equivalent structure metric few special cases these special cases briefly introduced brief remarks weyl conformal tensor weyl conformal theorem holonomy given