  We explore the implications of the requirement of Galilean invariance for
classical point particle lagrangians, in which the space is not assumed to be
flat to begin with. We show that for the free, time-independent lagrangian,
this requirement is equivalent to the existence of gradient Killing vectors on
space, which is in turn equivalent to the condition that the space is a direct
product, which is totally flat in the Galilean invariant direction. We then
consider more general cases and see that there is no simple generalisation to
these cases.
