  We develop a continuous theory of low-frequency dynamics for nanotubes with
truly two-dimensional (2D) walls constituted by single-atom monolayer. In this
frame topological bending elasticity of the monolayer is not related to its
vanishing macroscopic thickness. The proposed approach predicts completely new
sound dispersions and radius dependences of non-resonant Raman-active modes
frequencies in single-walled carbon nanotubes (SWCNT). Resulting relations are
suitable for nanotubes identification and more complete or alternative
characterization. The theory is also applied to describe the low-frequency
dynamics of double-walled carbon nanotubes (DWCNT). It establishes a clear-cut
relation between the radial breathing mode in SWCNT and breathing-like modes in
DWCNT and fits the existing Raman data better than previously developed 3D
continuous or discrete models. The results obtained constitute the basis for
new quantitative studies of the low-frequency vibrational spectrum, heat
capacity and heat transfer properties of carbon nanotubes.
