  Ricci and sectional curvatures of twisted flux tubes in Riemannian manifold
are computed to investigate the stability of the tubes. The geodesic equations
are used to show that in the case of thick tubes, the curvature of planar
(Frenet torsion-free) tubes have the effect ct of damping the flow speed along
the tube. Stability of geodesic flows in the Riemannian twisted thin tubes
(almost filaments), against constant radial perturbations is investigated by
using the method of negative sectional curvature for unstable flows. No special
form of the flow like Beltrami flows is admitted, and the proof is general for
the case of thin tubes. It is found that for positive perturbations and angular
speed of the flow, instability is achieved, since the sectional Ricci curvature
of the twisted tube metric is negative.
