Pullback: Encyclopedia II - Pullback - Pullback on tensors

Pullback - Pullback on tensors

Let be a linear map between vector spaces V and W. Then given a tensor T of rank (0,n) on W, another tensor, the pullback f ^* T on V can be defined. That is, given a tensor

and a set of vectors

one then defines the pullback as

The result f ^* T is again a tensor, so that f ^* is in fact a mapping from tensors on W to tensors on V. As a special case, note that if T is a (0,1)-tensor, so that , the dual space of W, then , and so the pullback acts in the reversed direction:

For a general f, a pullback can only be defined on tensors of rank (0,n). This is precisely because a pullback on mixed tensors would need to be "going in the opposed direction" for the contravariant indeces. If f is invertible, then this can be done, and one can define the pullback of an arbitrary mixed-rank tensor, as shown next. Let f be a linear isomorphism, so that is invertible. The pullback of a tensor of rank (n,0) can be defined by employing ; we make use of the identity (f ^{− 1}) ^* = (f ^* ) ^{− 1}. One then defines

Thus we've shown that if a linear transformation is invertible, it can be used to define the pullback on general tensors of mixed rank (m,n). Perhaps the easiest way to visualize and understand the above is to keep firmly in mind that f is nothing more than a matrix, so that f(v) is just the multiplication of a vector by a matrix. Similarly, the dual space should be visualized as nothing more than a dot product.

Adapted from the Wikipedia article "Pullback on tensors", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki