View source for Differential comitant

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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|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.