Abstract & Overview
The detailed description of the construction of the canonical form on the higher order frame bundle over an n-dimensional smooth manifold is given. In particular, it is shown that some vector space isomorphism playing the key role in this construction is defined correctly, i. e. it depends only on the frame of order p + 1 and does not depend on the choice of its representative, i. e. a local diffeomorphism which (p + 1)-jet is exactly this frame. This isomorphism acts from the direct sum of n-dimensional arithmetic space and the Lie algebra of the p-th order differential group to the tangent space to the p-th order frame bundle over the manifold at the p-th order frame lying “below”. The action of this isomorphism can be splitted into two its restrictions. The first one acts from the first direct summand, and the second one acts from the second direct summand. It is shown that the first restriction depends only on the choice of the (p + 1)-frame, while the second one is closely related to fundamental vector fields and therefore does not depend of this frame at all.