A difference approximation of the covariant derivative and other operators and geometric objects given in a Riemannian domain

Difference approximations for covariant derivatives of tensor fields of arbitrary rank given in Riemannian domain, retaining main geometrical properties, are suggested. An approach to its construction is based on certain difference analogs for Christoffel symbols. The main criterium here is exact vanishing for a difference covariant derivative of fundamental tensor and, in addition, an exact approximation of commutation relations which is possible only at a certain developed in the paper difference approximations for the curvature tensor.