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. The second
example indicates that for metrics of higher signature non-uniqueness need not
result from the existence of a de Rham decomposition or parallel null vector
fields.
