Discrete differential geometry. Consistency as integrability

Abstract

A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with finite number of elements (such as polyhedra), the discrete differential geometry aims at the development of discrete equivalents of notions and methods of smooth surface theory. Current interest in this field derives not only from its importance in pure mathematics but also from its relevance for other fields like computer graphics. Recent progress in discrete differential geometry has lead, somewhat unexpectedly, to a better understanding of some fundamental structures lying in the basis of the classical differential geometry and of the theory of integrable systems. The goal of this book is to give a systematic presentation of current achievements in this field.

A preliminary version of a book. 157 pp; See http://www.ams.org/bookstore-getitem/item=GSM-%5C98 for the final version appeared as: A.I. Bobenko, Yu.B. Suris. Discrete Differential Geometry. Integrable Structure. Graduate Studies in Mathematics, Vol. 98. AMS, 2008

References in corpus (3)

• Rhombic embeddings of planar graphs with faces of degree 4
• Exponentials form a basis of discrete holomorphic functions
• Integrable lattices and their sub-lattices: from the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme

Cited by in corpus (17)

• On organizing principles of Discrete Differential Geometry. Geometry of spheres
• The B-quadrilateral lattice, its transformations and the algebro-geometric construction
• Isothermic surfaces in sphere geometries as Moutard nets
• On vector analogs of the modified Volterra lattice
• Classification of 3-dimensional integrable scalar discrete equations
• Discrete complex analysis on planar quad-graphs
• Geometric discrete analogues of tangent bundles and constrained Lagrangian systems
• Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency
• On discrete integrable equations with convex variational principles
• Classical integrable field theories in discrete 2+1 dimensional space-time
• Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations
• Geometry of quadrilateral nets: second Hamiltonian form
• Discrete Affine Minimal Surfaces with Indefinite Metric
• Pseudospherical surfaces on time scales
• Discrete Affine Surfaces based on Quadrangular Meshes
• Bianchi surfaces. Integrability in arbitrary parametrization
• Asymptotic Nets and Discrete Affine Surfaces with Indefinite Metric