A Discrete Extrinsic and Intrinsic Dirac Operator

Abstract

In differential geometry of surfaces the Dirac operator appears intrinsically as a tool to address the immersion problem as well as in an extrinsic flavor (that comes with spin transformations to comformally transfrom immersions) and the two are naturally related. In this paper we consider a corresponding pair of discrete Dirac operators, the latter on discrete surfaces with polygonal faces and normals defined on each face, and show that many key properties of the smooth theory are preserved. In particular, the corresponding spin transformations, conformal invariants for them, and the relation between this operator and its intrinsic counterpart are discussed.

Keywords:

• Discrete differential geometry
• spin geometry
• surface theory

Declaration of interest

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by the DFG-Collaborative Research Center, TRR 109,” Discretization in Geometry and Dynamics”