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Since electrons in a ballistic regime perceive a carbon nanotube or agraphene layer structure as a continuous medium, we can use the study of thequantum dynamics of one electron constrained to a curve or surface to obtain aqualitative description of the conduction electrons' behavior. The confinementprocess of a quantum particle to a curve or surface leads us, in the so-called"confining potential formalism" (CPF), to a geometry-induced potential (GIP) inthe effective Schr\"odinger equation. With these considerations, this work aimsto study in detail the consequences of constraining a quantum particle to ahelix, catenary, helicoid, or catenoid, exploring the relations between thesecurves and surfaces using differential geometry. Initially, we use thevariational method to estimate the energy of the particle in its ground state,and thus, we obtain better approximations with the use of the confluent Heunfunction through numerical calculations. Thus, we conclude that a quantumparticle constrained to an infinite helix has its angular momentum quantizeddue 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 inexcited states which constitute a continuous energy band. Additionally, wepropose measurements of physical observables capable of discriminating thetopologies of the studied surfaces, in the context of topological metrology.

Geometric effects on the electronic structure of curved nanotubes and curved graphene: the case of the helix, catenary, helicoid, and catenoid