  Swimming in curved spacetime is a phenomenon in which free bodies in curved
spacetimes are able to propel themselves by performing cyclic internal motions.
Most surprisingly, it has been suggested that, in the limit of fast cycles, the
effective motion would not depend on the rapidity of the internal motions but
only on the sequence of shapes assumed by the body, just like a swimmer in a
non-turbulent viscous fluid. Since a rigorous treatment of this problem was
still missing, in this paper we present a general, covariant theory of swimming
in curved spacetime. We show that the effect can indeed take place, but it only
displays this geometric-phase character in special circumstances (even within
the limit of fast cycles). Our methods are fairly general, providing a
technique to study the motion of free small articulated bodies in general
relativity by mapping the problem to an analogue in special relativity.
