# Geometric influences of a particle confined to a curved surface embedded in three-dimensional Euclidean space

## TL;DR

This paper derives formulas for geometric effects on a particle confined to a curved surface in 3D space, revealing how curvature influences quantum properties like potential, momentum, and spin-orbit couplings.

## Contribution

It introduces a comprehensive formula for geometric influences on particles on curved surfaces, including new insights into geometric spin-orbit interactions and their dependence on surface geometry.

## Key findings

- Geometric orbital angular momentum can induce azimuthal spin polarization.
- Sign of geometric Dresselhaus coupling depends on surface inclination angle.
- Derived formulas applicable to various curved surface quantum systems.

## Abstract

In the spirit of the thin-layer quantization approach, we give the formula of the geometric influences of a particle confined to a curved surface embedded in three-dimensional Euclidean space. The geometric contributions can result from the reduced commutation relation between the acted function depending on normal variable and the normal derivative. According to the formula, we obtain the geometric potential, geometric momentum, geometric orbital angular momentum, geometric linear Rashba and cubic Dresselhaus spin-orbit couplings. As an example, a truncated cone surface is considered. We find that the geometric orbital angular momentum can provide an azimuthal polarization for spin, and the sign of the geometric Dresselhaus spin-orbit coupling can be flipped through the inclination angle of generatrix.