ar X iv : m at h - ph / 0 20 80 41 v 1 2 9 A ug 2 00 2 Geometry of the Triangle Equation on Two - Manifolds ∗

A non-traditional approach to the discretization of differential-geometrical connections was suggested by the authors in 1997. At the same time we started studying first order difference “black and white triangle operators (equations)” on triangulated surfaces with a black and white coloring of triangles. In this work, we develop the theory of these operators and equations, showing their similarity with the complex derivatives ∂ and ∂. Introduction Do there exist any natural difference analogs of the operators ∂ and ∂ on the complex plane? No theory of difference “first order” operators that have some properties similar to those of ∂ and ∂ has been developed before; the attention of the literature is paid only to discrete analogs of the Laplace– Beltrami operator, which are “second order” operators. However, in papers [1, 2, 3] we already studied some “first order” difference operators, starting The work of I. Dynnikov is partially supported by Russian Foundation for Fundamental Research, grant no. 02-01-00659, and by Leading Scientific School Support grant no. 0015-96011; the work of S. Novikov is supported by NSF (USA), grant no. DMS-00772700 Mech. and Math. Dept., Moscow State University, Moscow 119992 GSP-2, Russia, e-mail: [email protected] Institute for Physical Sciences and Technology, Univ. of Maryland at College Park, MD 20742, USA; L.D. Landau Institute for Theoretical Physics, Kosygyna str. 2, Moscow 117940, Russia, e-mail: [email protected]