On the Spinor Representation of Surfaces in Euclidean 3-Space

TL;DR

This paper explores how surfaces in Euclidean 3-space can be represented using spinor fields, linking surface immersions to solutions of the inhomogeneous Dirac equation, thus providing a spinor-based framework for surface geometry.

Contribution

It establishes a clear relationship between surface immersions and solutions to the inhomogeneous Dirac equation using spinor fields, offering a novel perspective in differential geometry.

Findings

Restriction of parallel spinors yields solutions to the Dirac equation on surfaces.

Surface geometry can be characterized via spinor fields satisfying the Dirac equation.

Provides a spinor-based description of surface immersions in Euclidean space.

Abstract

The aim of the present paper is to clarify the relationship between immersions of surfaces and solutions of the inhomogeneous Dirac equation. The main idea leading to the description of a surface M^2 by a spinor field is the observation that the restriction to M^2 of any parallel spinor phi on R^3 is (with respect to the inner geometry of M^2) a non-trivial spinor field on M^2 of constant length which is a solution of the inhomogeneous Dirac equation and vice versa.