On construction of the canonical form on the frame bundle

Abstract

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.

Keywords

• smooth manifold
• jet
• frame bundle
• canonical form