Difference discrete connection and curvature on cubic lattice

Abstract

In a way similar to the continuous case formally, we define in different but equivalent manners the difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space. We deal with the difference operators as the discrete counterparts of the derivatives based upon the differential calculus on the lattice. One of the definitions can be extended to the case over the random lattice. We also discuss the relation between our approach and the lattice gauge theory and apply to the discrete integrable systems.

Exterior Difference System on Hypercubic Lattice

“…At first, we briefly recall the definitions and propositions of discrete tangent bundle, discrete cotangent bundle and discrete exterior derivative operator [9].…”

“…Proposition 2. 9 Cartan lemma in discrete case: Let V be a vector space and suppose there is a quadratic relation…”

“…Within a framework of noncommutative geometry, many discrete analogous of the concepts of differential geometry and Riemannian geometry have been considered by many authors [3][4][5][6][7][8][9][10][11][12][13]. Recently, a systematic manner about how to get the discrete counterparts from continuous system was given by K. Wu et al [9].…”

“…Recently, a systematic manner about how to get the discrete counterparts from continuous system was given by K. Wu et al [9]. One of the key points in their approach is still to regard the difference operators acting on the functions space over the regular lattice as a kind of independent geometric objects.…”

Formal Integrability Criteria for Nonlinear Partial Difference Equations

“…Some basic concepts in formal differential geometry such as Jet bundle and exterior differential systems also have been discretized and have many applications in discrete mechanics and difference equations [10][11][12][13][14][15][16][17]. The main result of formal differential geometry is to determine when a partial differential equations can be extended by suitable compatability conditions, meaning that the extended system has formal power series solutions.…”