  Since electrons in a ballistic regime perceive a carbon nanotube or a
graphene layer structure as a continuous medium, we can use the study of the
quantum dynamics of one electron constrained to a curve or surface to obtain a
qualitative description of the conduction electrons' behavior. The confinement
process of a quantum particle to a curve or surface leads us, in the so-called
da Costa's approach, to a geometry-induced potential (GIP) in the effective
Schr\"odinger equation. With these considerations, this work aims to study in
detail the consequences of constraining a quantum particle to a helix,
catenary, helicoid, or catenoid, exploring the relations between these curves
and surfaces using differential geometry. Initially, we use the variational
method to estimate the energy of the particle in its ground state, and then, we
obtain better approximations with the use of the confluent Heun function
through numerical calculations. Thus, we conclude that a quantum particle
constrained to an infinite helix has its angular momentum quantized due to the
geometry of the curve, while in the cases of the catenary, helicoid, and
catenoid the particle can be found either in a single bound state or in excited
states which constitute a continuous energy band.
