Invariants of Immersions on n-Dimensional Affine Manifold

Main results: The system of Christoffel symbols of the connection of an immersion 4.: 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 4 . , 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 4.: 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.