We consider the Laplace operator in a thin three dimensional tube with a
Robin type condition on its boundary and study, asymptotically, the spectrum of
such operator as the diameter of the tube's cross section becomes
infinitesimal. In contrast with the Dirichlet condition case, we evidence
different behaviors depending on a symmetry criterium for the fundamental mode
in the cross section. If that symmetry condition fails, then we prove the
localization of lower energy levels in the vicinity of the minimum point of a
suitable function on the tube's axis depending on the curvature and the
rotation angle. In the symmetric case, the behavior of lower energy modes is
shown to be ruled by a one dimensional Sturm-Liouville problem involving an
effective potential given in explicit form
