On the spinor representation of surfaces in Euclidean 3-space

✍ Scribed by Thomas Friedrich

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 817 KB
• Volume: 28
• Category: Article
• ISSN: 0393-0440
• DOI: 10.1016/s0393-0440(98)00018-7

✦ Synopsis

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 M2 by a spinor field is the observation that the restriction to M2 of any parallel spinor $ on Iw3 is a non-trivial spinor field on M2 of constant length which is a solution of the inhomogeneous Dirac equation. Vice versa, any solution of the equation O($) = H . p+ of constant length defines a symmetric endomorphism satisfying the Gauss-and Codazzi equations, i.e. an isometric immersion of M2 into the 3-dimensional Euclidean space.