Discrete connections on the triangulated manifolds and difference linear equations
Abstract
Following the previous authors works (joint with I.A.Dynnikov) we develop a theory of the discrete analogs of the differentialgeometrical (DG) connections in the triangulated manifolds. We study a nonstandard discretization based on the interpretation of DG Connection as linear first order (''triangle'') difference equation acting on the scalar functions of vertices in any simplicial manifold. This theory appeared as a byproduct of the new type of discretization of the special Completely Integrable Systems, such as the famous 2D Toda Lattice and corresponding 2D stationary Schrodinger operators. A nonstandard discretization of the 2D Complex Analysis based on these ideas was developed in our recent work closely connected with this one. A complete classification theory is constructed here for the Discrete DG Connections based on the mixture of the abelian and nonabelian features.
 Publication:

arXiv eprints
 Pub Date:
 March 2003
 DOI:
 10.48550/arXiv.mathph/0303035
 arXiv:
 arXiv:mathph/0303035
 Bibcode:
 2003math.ph...3035N
 Keywords:

 Mathematical Physics;
 Mathematics  Algebraic Topology;
 Mathematics  Differential Geometry;
 Mathematics  Mathematical Physics
 EPrint:
 Latex 17 pages, several technical mistakes are corrected, complete solution of the reconstruction problem for higher dimensions is added