Surface theory in discrete projective differential geometry. I. A canonical frame and an integrable discrete Demoulin system. (30th June 2018)

• Record Type: Journal Article
• Authors: Schief, W. K.; Szereszewski, A.
• Is Part Of: Proceedings. Volume 474:Number 2214(2018)
• Publication Date: 2018-06-30
• Subjects: projective differential geometry -- discrete differential geometry -- integrable system -- Bäcklund transformation; Physical sciences -- Periodicals; Engineering -- Periodicals; Mathematics -- Periodicals; 500
• DOI: 10.1098/rspa.2017.0770
• Languages: English
• ISSNs: 1364-5021

Abstract

Abstract : We present the first steps of a procedure which discretizes surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi equations adopt compact forms. Based on a scaling symmetry which injects a parameter into the linear Gauss-Weingarten equations, we set down an algebraic classification scheme of discrete projective minimal surfaces which turns out to admit a geometric counterpart formulated in terms of discrete notions of Lie quadrics and their envelopes. In the case of discrete Demoulin surfaces, we derive a Bäcklund transformation for the underlying discrete Demoulin system and show how the latter may be formulated as a two-component generalization of the integrable discrete Tzitzéica equation which has originally been derived in a different context. At the geometric level, this connection leads to the retrieval of the standard discretization of affine spheres in affine differential geometry.