Abstract
Main results: The system of Christoffel symbols of the connection of an immersion ξ:J→R^n of an n-dimensional manifold J in the n-dimensional linear space R^n is a system of generators of the differential field of all Aff(n)-invariant differential rational functions of ξ, where Aff(n) is the group of all affine transformations of R^n. A similar result have obtained for the subgroup SAff(n) of Aff(n) generated by all unimodular linear transformations and parallel translations of R^n. Rigidity and uniqueness theorems for immersions ξ:J→R^n in geometries of groups Aff(n) and SAff(n) were obtained. These theorems are given in terms of the affine connection and the volume form of immersions.
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