  We study a free quantum motion on periodically structured manifolds composed
of elementary two-dimensional "cells" connected either by linear segments or
through points where the two cells touch. The general theory is illustrated
with numerous examples in which the elementary components are spherical
surfaces arranged into chains in a straight or zigzag way, or two-dimensional
square-lattice "carpets". We show that the spectra of such systems have an
infinite number of gaps and that the latter dominate the spectrum at high
energies.
