The Lie derivative as a canonical morphism

Abstract

First, there is a proper definition of a canonical morphism for a given principal G bundle over some smooth manifold. A complete description is given of all canonical morphisms attached to the linear frame bundle (which involves frames having first-order contact with the manifold). Second, the Lie derivative is shown to be a canonical morphism for the quadratic frame bundle (this comprises frames with second-order contact). As a practical application, one obtains an explicit, coordinate-free description of the Lie derivative on any tensor bundle. Simultaneously, there appears a natural method of generating connections on any tensor bundle from a given connection on the tangent bundle.