  We investigate the combined effects of spatial curvature and topology on the
properties of the vacuum state for a charged scalar field localized on
rotationally symmetric 2D curved tubes. For a general spatial geometry and for
quasiperiodicity condition with a general phase, the representation of the
Hadamard function is provided where the topological contribution is explicitly
extracted. As an important local characteristic of the vacuum state the
expectation value of the current density is studied. The vacuum current is a
periodic function of the magnetic flux enclosed by the tube with the period of
flux quantum. The general formula is specified for constant radius and conical
tubes. As another application, we consider the Hadamard function and the vacuum
current density for a scalar field on the Beltrami pseudosphere. Several
representations are provided for the corresponding expectation value. For small
values of the proper radius of the tube, compared with the curvature radius,
the effect of spatial curvature on the vacuum current is weak and the leading
term in the corresponding expansion coincides with the current density on a
constant radius tube. The effect of curvature is essential for proper radii of
the tube larger than the radius of spatial curvature. In this limit the
fall-off of the current density, as a function of the proper radius, follows a
power-law for both massless and massive fields. This behavior is in clear
contrast to the one for a constant radius tube with exponential decay for
massive fields. We also compare the vacuum currents on the Beltrami
pseudosphere and on locally de Sitter and anti-de Sitter 2D tubes.
