Differentical equations, dynamical systems and optimal control

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

E. Yu. Derevtsov<sup>ab

a</sup> Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia
^b Novosibirsk state university, Pirogova st., 2, 630090, Novosibirsk, Russia

Abstract: 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.

Keywords: difference approximation, Riemannian domain, covariant derivative, tensor of curvature, Christoffel symbols, Ricci formulae.

UDC: 517.9

MSC: 65D25

Received August 30, 2015, published December 15, 2015

DOI: 10.17377/semi.2015.12.084