Recurrence structures in 4-dimensional manifolds with metric of signature (+,+,−,−)

Abstract

This paper studies the recurrence structure of second order symmetric tensors on a 4 -dimensional manifold M admitting a metric g of neutral signature ( + , + , − , − ) . The technique used is to first solve the problem when the tensor in question is either parallel (covariantly constant) or can be scaled so that it is parallel. Then one considers those recurrent tensors which are not in this category. The general approach is based on a (known) classification scheme for second order symmetric tensors (which will be reviewed) and a method of distinguishing those tensor types which may be (up to a scaling) parallel and those which may not be. The analysis is based on the holonomy group of (the Levi-Civita connection associated with) g , the possible Lie algebras of which are known. The results are then applied to the Ricci tensor of the structure ( M , g ) (Ricci-recurrence), including the study of those holonomy types which permit Einstein spaces, and also to the problem of finding the complete set of metrics associated with a particular Levi-Civita connection of ( M , g ) . Some of the results hold quite generally for any dimension of M and signature of g and this will be used to (very briefly) consider the same problem for Lorentz and positive definite signature (and whose solution is known). A by-product of this study is the finding of certain useful, direct techniques to study the properties of the various holonomy types using direct algebraic methods and the Ambrose–Singer theorem.