  We theoretically show that the topology of a non-simply-connected annular
atomic Bose-Einstein condensate enforces the inner surface waves to be always
excited with outer surface excitations and that the inner surface modes are
associated with induced vortex dipoles unlike the surface waves of a
simply-connected one with vortex monopoles. Consequently, under stirring to
drive an inner surface wave, a peculiar population oscillation between the
inner and outer surface is generated regardless of annulus thickness. Moreover,
a new vortex nucleation process by stirring is observed that can merge the
inner vortex dipoles and outer vortex into a single vortex inside the annulus.
The energy spectrum for a rotating annular condensate with a vortex at the
center also reveals the distinct connection of the Tkachenko modes of a vortex
lattice to its inner surface excitations.
