A pullback is a general categorical operation appearing in a number of mathematical contexts, sometimes going under a different name. If
is a linear transformation between
vector spaces, then 
(usually called transpose map or dual map because
its associated matrix is the transpose of 
) is an example of a pullback map.
In the case of a diffeomorphism and smooth manifold, a very explicit definition can be formulated. Given an 
-form 
on a manifold 
, define the 
-form 
on 
by its action on an 
-tuple of tangent vectors 
as the number 
. This defines a
map on 
-forms
and is the pullback map.
