  We investigate the low-energy properties of N-leg integer-spin ladders and
tubes with an antiferromagnetic (AF) intrachain coupling. In the odd-leg tubes,
the AF rung coupling causes the frustration. To treat all ladders and tubes
systematically, we apply S\'en\'echal's method [Phys. Rev. B 52, 15319 (1995)],
based on the nonlinear sigma model. This strategy is valid in the weak
interchain (rung) coupling regime. We show that all frustrated tubes possess
six-fold degenerate spin-1 magnon bands, as the lowest excitations, while other
ladders and tubes have a standard triply degenerate bands. We also consider
effects of four kinds of Zeeman terms: uniform, staggered only along the rung,
only along the chain, or both directions. The above prediction of the no-field
case implies that a sufficiently strong uniform field yields a two-component
Tomonaga-Luttinger liquid (TLL) due to the condensation of doubly degenerate
lowest magnons in frustrated tubes. In contrast, the field induces a standard
one-component TLL in all other systems. This is supported by symmetry and
bosonization arguments based on the Ginzburg-Landau theory. The bosonization
also suggests that the two-component TLL vanishes and a one-component TLL
appears, when the uniform field becomes larger for the second lowest magnon
bands to touch the zero-energy line. This transition could be observed as a
cusp singularity in the magnetization process. All the analyses for the systems
with a staggered Zeeman term suggest that the emergence of the doubly
degenerate transverse magnons and the single longitudinal one is universal for
the one-dimensional AF spin systems with weak staggered field.
