  Context. Fast rotating red giants in the upper part of the red giant branch
have surface velocities that cannot be explained by single star evolution.
  Aims. We check whether tides between a star and a planet followed by planet
engulfment can indeed accelerate the surface rotation of red giants for a
sufficient long time in order to produce these fast rotating red giants.
  Methods. Using rotating stellar models, accounting for the redistribution of
the angular momentum inside the star by different transport mechanisms, for the
exchanges of angular momentum between the planet orbit and the star before the
engulfment and for the deposition of angular momentum inside the star at the
engulfment, we study how the surface rotation velocity at the stellar surface
evolves.
  Results. We show that the surface velocities reached at the end of the
orbital decay due to tidal forces and planet engulfment can be similar to
values observed for fast rotating red giants. This surface velocity then
decreases when the star evolves along the red giant branch but at a
sufficiently slow pace for allowing stars to be detected with such a high
velocity. More quantitatively, star-planet interaction can produce a rapid
acceleration of the surface of the star, above values equal to 8 km s$^{-1}$,
for periods lasting up to more than 30% the red giant branch phase. The changes
of the surface carbon isotopic ratios produced by the dilution of the planetary
material into the convective envelope is quite modest. Much more important
might be the increase of the lithium abundance due to this effect. However
lithium may be affected by many different, still uncertain, processes.
  Conclusions. The acceleration of the stellar surface to rotation velocities
above limits that depend on the surface gravity does appear at the moment as
the clearest signature of a star-planet interaction.[abridged]
