  This thesis is devoted to the Differential Geometry of curves and surfaces
along with applications in Quantum Mechanics. In the 1st part we introduce the
well known Frenet frame. Later, we show that the curvature function is a lower
bound for the scalar angular velocity of any other orthonormal moving frame,
from which one defines Rotation Minimizing (RM) frames as the ones that achieve
this minimum. Remarkably, RM frames are ideal to study spherical curves and
allow us to characterize them through a simple linear equation. We also apply
these ideas to curves that lie on level surfaces, by reinterpreting the problem
in the context of a metric induced by a Hessian, which may fail to be positive
or non-degenerate and naturally leads us to a Lorentz-Minkowski
$\mathbb{E}_1^3$ or isotropic $\mathbb{I}^3$ space. Here we develop a
systematic approach to construct RM frames and characterize spherical curves in
$\mathbb{E}_1^3$ and $\mathbb{I}^3$ and furnish a criterion for a curve to lie
on a level surface. Finally, we extend these tools to characterize geodesic
spherical curves in hyperbolic and spherical spaces. In the 2nd part we apply
the previous ideas in the quantum dynamics of a constrained particle. After
describing the confining potential formalism, from which emerges a
geometry-induced potential (GIP), we devote our attention to tubular surfaces
to model curved nanotubes. The use of RM frames offers a simpler description
for the constrained dynamics and allows us to show that the torsion of the
centerline of a tube gives rise to a geometric phase. Later, we study the
problem of prescribed GIP for surfaces. Here we explore the GIP for surfaces
invariant by a 1-parameter group of isometries, i.e., cylindrical, revolution,
and helicoidal surfaces. Finally, we devote a special attention to helicoidal
minimal surfaces and prove the existence of geometry-induced bound and
localized states.
