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 and then discuss plane curves with a power-law curvature. Later, we show that the curvature function is a lower bound for the scalar angular velocity of any moving frame, from which one defines Rotation Minimizing (RM) frames as those frames that achieve this minimum. Remarkably, RM frames are ideal for studying spherical curves and allow us to characterize them through a linear equation, contrasting with a differential equation from a Frenet approach. We also apply these ideas to curves that lie on level surfaces of a smooth function F by reinterpreting the problem in the context of a metric induced by Hess F, which may fail to be positive or non-degenerate and naturally leads us to a Lorentz-Minkowski or isotropic space. We then develop a systematic approach to construct RM frames, characterize spherical curves, and furnish a criterion for a curve to lie on a level set surface. Finally, we extend these investigations to characterize curves on the (hyper)surface of geodesic spheres in a Riemannian manifold. Since the (radial) geodesics connecting geodesic spherical curves to a fixed point induce a normal vector field, we can characterize geodesic spherical curves in hyperbolic and spherical geometries through a linear equation. In the 2nd part, we apply some of the previous ideas in the quantum dynamics of a constrained particle, where differential geometry is a relevant timing tool due to the possibility of synthesizing nanostructures with non-trivial shapes. After describing the confining potential formalism from which a geometry-induced potential (GIP) emerges, we devote our attention to tubular surfaces to model curved nanotubes. The use of RM frames offers a simpler description of the constrained dynamics and shows that the torsion of the centerline of a curved tube gives rise to a geometric phase. Later, we study the problem of prescribed GIP for curves and surfaces in Euclidean space: for curves, it is solved by integrating Frenet equations, while for surfaces, it involves a non-linear 2nd-order PDE. Here, we explore the GIP for surfaces invariant by a 1-parameter group of isometries, which turns the PDE into an ODE and leads to cylindrical, revolution, and helicoidal surfaces. The latter class is an important candidate to establish a link with chirality. We devote special attention to helicoidal minimal surfaces and prove the existence of geometry-induced bound and localized states and the possibility of controlling the change in the probability density when the surface is subjected to an extra charge.