Differential Geometry of Rotation Minimizing Frames, Spherical Curves, and Quantum Mechanics of a Constrained Particle

Abstract

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.