On the spinor representation of surfaces in Euclidean 3-space

Abstract

The aim of the present paper is to clarify the relationship between immersions of surfaces and solutions of the 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 Ψ on R 2 is a non-trivial spinor field on M 2 of constant length which is a solution of the inhomogeneous Dirac equation. Vice versa, any solution of the equation D( Ψ) = H · Ψ of constant length defines a symmetric endomorphism satisfying the Gauss-and Codazzi equations, i.e. an isometric immersion of M 2 into the 3-dimensional Euclidean space.