Non-uniqueness of metrics compatible with a symmetric connection

Abstract

In Riemannian geometry the only ambiguity in a metric that is compatible with a symmetric connection arises from scaling by constants and de Rham decompositions. The situation for pseudo-Riemannian metrics is, however, quite different. In this paper this phenomenon is studied. One of the main conclusions reached is that there is an intimate link between the problem of the existence of alternative metrics for a symmetric connection of higher order tangent bundle structures. Such alternative metrics are provides a solution to the problem in principle, the difficulties of which derive from the plethora of Jordan canonical forms of a nilpotent matrix. The paper concludes with a description of all alternative metrics for manifolds of dimension five and less.