Differential comitant

A differentiable mapping $ \phi $ of a tensor bundle $ T $ on a manifold $ M $ into a tensor bundle $ T ^ { \prime } $ on the same manifold such that if $ p : T \rightarrow M $ and $ p ^ \prime : T ^ { \prime } \rightarrow M $ are the projections of $ T $ and $ T ^ { \prime } $ on $ M $, then

$$ p ^ \prime \phi = p . $$

The components of the tensor $ T ^ { \prime } = \phi ( T) $ in a local chart $ \xi $ on $ M $ depend on $ \xi $ only by means of the components of the tensor $ T $.

In particular, when $ T ^ { \prime } $ is reduced to the bundle of relative scalars of weight $ g $, the differential comitant is a differential invariant of weight $ g $.

Comments

Thus, a differential comitant is simply a vector bundle mapping from the tensor bundle $ T $ to the tensor bundle $ T ^ { \prime } $.

The bundle of relative scalars of weight $ g $ is constructed as follows. It is a line bundle. Let $ ( U _ \alpha ) _ \alpha $ be an atlas for the manifold $ M $ with coordinate change diffeomorphisms $ \phi _ {\alpha \beta } : U _ {\alpha \beta } \rightarrow U _ {\beta \alpha } $. Take the trivial line bundles $ U _ \alpha \times \mathbf R $ over each $ U _ \alpha $ and glue them together by means of the diffeomorphisms $ \widetilde \phi _ {\alpha \beta } : U _ {\alpha \beta } \times \mathbf R \rightarrow U _ {\beta \alpha } \times \mathbf R $, $ ( x , t ) \mapsto ( \phi _ {\alpha \beta } ( x ) , ( \mathop{\rm det} ( J ( \phi _ {\alpha \beta } ) ( x) )) ^ {g} t ) $, where $ J ( \phi _ {\alpha \beta } ) ( x) : T _ {x} U _ {\alpha \beta } \rightarrow T _ {\phi _ {\alpha \beta } ( x) } U _ {\beta \alpha } $ is the Jacobian matrix of $ \phi _ {\alpha \beta } $ at $ x $.

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.