Nonuniqueness of the metric in Lorentzian manifolds

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This paper is concerned with the correspondence between a Lorentzian metric and its Levi-Civita connection. Although each metric determines a unique compatible symmetric connection, it is possible for more than one metric to engender the same connection. This non-uniqueness is studied for metrics of arbitrary signature and for Lorentzian metrics is shown to arise either from a de Rham-Wu decomposition or a local parallel null vector field. A key ingredient in the analysis is the construct of a submersive connection in which a connection passes to a quotient space. Finally, two examples of metrics are given, the first of which shows that the metric may be non-unique even though a null vector field exists only locally