Abstract
Motivated by the equivariant moving frame method and starting from the study about the prolongation of the infinitesimal generators of transformation groups and the universal recurrence formulae for differential invariants, we present some theories which make it possible for the systematic determination of the higher order prolongation of the infinitesimal generators. Furthermore, based on the symbolic computation and the fundamental sets of the determined normalized differential invariants, we give the exact recurrence formulae to construct the infinitely many higher order normalized differential invariants and syzygies which show the relationships between them. In addition, by the obtained normalized differential invariants, the explicit Monge-Taylor forms are obtained extensively for submanifolds under group actions.
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