Geodesic
Discrete Connections and Difference Linear Equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology and set theory, Tome 247 (2004), pp. 186-201.

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Following earlier works, we develop here a nonstandard discrete analogue of the theory of differential-geometric $GL_{n}$-connections on triangulated manifolds. This theory is based on the interpretation of a connection as a first-order linear difference equation—the “triangle equation”—for scalar functions of vertices in simplicial complexes. This theory appeared as a byproduct of the discretization of famous completely integrable systems such as the 2D Toda lattice. A nonstandard discretization of complex analysis based on these ideas was developed earlier. Here, a complete classification theory based on the mixture of abelian and nonabelian features is given for connections on triangulated manifolds. 
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S. P. Novikov. Discrete Connections and Difference Linear Equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology and set theory, Tome 247 (2004), pp. 186-201. http://geodesic.mathdoc.fr/item/TM_2004_247_a12/

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