  By means of {\it ab initio} simulations, we investigate the phonon band
structure and electron-phonon coupling in small 4-\AA diameter nanotubes. We
show that both the C(5,0) and C(3,3) tubes undergo above room temperature a
Peierls transition mediated by an acoustical long-wavelength and an optical
$q=2k_F$ phonons respectively. In the armchair geometry, we verify that the
electron-phonon coupling parameter $\lambda$ originates mainly from phonons at
$q=2k_F$ and is strongly enhanced when the diameter decreases. These results
question the origin of superconductivity in small diameter nanotubes.
