  We present a theoretical formulation of the one-electron problem constrained
on the surface of a cylindrical tubule with varying diameter. Because of the
cylindrical symmetry, we may reduce the problem to a one-dimensional equation
for each angular momentum quantum number $m$ along the cylindrical axis. The
geometrical properties of the surface determine the electronic structures
through the geometry dependent term in the equation. Magnetic fields parallel
to the axis can readily be incorporated. Our formulation is applied to simple
examples such as the catenoid and the sinusoidal tubules. The existence of
bound states as well as the band structures, which are induced geometrically,
for these surfaces are shown. To show that the electronic structures can be
altered significantly by applying a magnetic field, Aharonov-Bohm effects in
these examples are demonstrated.
