Surface theory in discrete projective differential geometry. I. A canonical frame and an integrable discrete Demoulin system

• Published in: Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Mathematical, physical, and engineering sciences, 2018-06, Vol.474 (2214), p.20170770-20170770
• Main Authors: Schief, W. K., Szereszewski, A.
• Format: Article
• Language: English
• Subjects: Bäcklund Transformation; Differential geometry; Discrete Differential Geometry; Geometry; Integrable System; Mathematical analysis; Minimal surfaces; Projective Differential Geometry
• ISSN: 1364-5021; 1471-2946
• DOI: 10.1098/rspa.2017.0770

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.