ON INVARIANTS OF IMMERSIONS OF AN n-DIMENSIONAL MANIFOLD IN AN n-DIMENSIONAL PSEUDO-EUCLIDEAN SPACE

Let En p be the n-dimensional pseudo-Euclidean space of index p and M(n, p) the group of all transformations of E(p)(n) generated by pseudo-orthogonal transformations and parallel translations. We describe the system of generators of the differential field of all M(n, p)-invariant differential rational functions of a map x : J -> E(p)(n) of an open subset J subset of E(p)(n). Using this result, we prove analogues of the Bonnet theorem for immersions of an n-dimensional C(infinity)-manifold J in E(p)(n). These analogues are given in terms of the pseudo-Riemannian metric, the volume form, and the connection on J induced by the immersion of J in E(p)(n).