Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds

TL;DR

This paper investigates the relationship between holonomy groups and projective equivalence in 4-dimensional Lorentz manifolds, providing new insights into their geometric structure and classifications.

Contribution

It establishes new connections between projective relatedness and holonomy types, including a comprehensive review of holonomy groups and curvature tensor classifications.

Findings

Holonomy types are linked to projective equivalence in Lorentz manifolds.

A classification of curvature tensors relevant to these manifolds is provided.

New results relate holonomy groups to geodesic structures.

Abstract

A study is made of 4-dimensional Lorentz manifolds which are projectively related, that is, whose Levi-Civita connections give rise to the same (unparameterised) geodesics. A brief review of some relevant recent work is provided and a list of new results connecting projective relatedness and the holonomy type of the Lorentz manifold in question is given. This necessitates a review of the possible holonomy groups for such manifolds which, in turn, requires a certain convenient classification of the associated curvature tensors. These reviews are provided.