Differential comitant

A differentiable mapping of a tensor bundle on a manifold into a tensor bundle on the same manifold such that if and are the projections of and on , then

The components of the tensor in a local chart on depend on only by means of the components of the tensor .

In particular, when is reduced to the bundle of relative scalars of weight , the differential comitant is a differential invariant of weight .

Comments

Thus, a differential comitant is simply a vector bundle mapping from the tensor bundle to the tensor bundle .

The bundle of relative scalars of weight is constructed as follows. It is a line bundle. Let be an atlas for the manifold with coordinate change diffeomorphisms . Take the trivial line bundles over each and glue them together by means of the diffeomorphisms , , where is the Jacobian matrix of at .

Cf. also Differential invariant. Note however that for differential invariants not only tensor bundles but also (tensor and exterior products of) higher jet bundles and their duals are considered.