  The Dirichlet Laplacian in curved tubes of arbitrary constant cross-section
rotating together with the Tang frame along a bounded curve in Euclidean spaces
of arbitrary dimension is investigated in the limit when the volume of the
cross-section diminishes. We show that spectral properties of the Laplacian
are, in this limit, approximated well by those of the sum of the Dirichlet
Laplacian in the cross-section and a one-dimensional Schrodinger operator whose
potential is expressed solely in terms of the first curvature of the reference
curve. In particular, we establish the convergence of eigenvalues, the uniform
convergence of eigenfunctions and locate the nodal set of the Dirichlet
Laplacian in the tube near nodal points of the one-dimensional Schrodinger
operator. As a consequence, we prove the "nodal-line conjecture" for a class of
non-convex and possibly multiply connected domains. The results are based on a
perturbation theory developed for Schrodinger-type operators in a straight tube
of diminishing cross-section.
