  It has been argued that an extended, quasi-rigid body evolving freely in
curved spacetime can deviate from its natural trajectory by simply performing
cyclic deformations. More interestingly, in the limit of rapid cycles, the
amount of deviation, per cycle, would depend on the sequence of deformations
but not on how fast they are performed -- like the motion of a swimmer at low
Reynolds number. Here, however, we show that the original analysis which
supported this idea is inappropriate to investigate the motion of extended
bodies in the context of general relativity, rendering its quantitative results
invalid and casting doubts on the reality of this swimming effect. We
illustrate this by showing that the original analysis leads to a non-zero
deviation even in a scenario where no swimming can possibly occur.
Notwithstanding, by applying a fully covariant, local formalism, we show that
swimming in curved spacetime is indeed possible and that, in general, its
magnitude can be of the same order as (fortuitously) anticipated -- although it
is highly suppressed in the particular scenario where it was originally
investigated.
