Pullback Map

A pullback is a general Categorical operation appearing in a number of mathematical contexts, sometimes going under a different name. If $T:V\rightarrow W$ is a linear transformation between Vector Spaces, then $T^*: W^*\rightarrow V^*$ (usually called Transpose Map or Dual Map because its associated matrix is the Matrix Transpose of $T$) is an example of a pullback map.

In the case of a Diffeomorphism and Differentiable Manifold, a very explicit definition can be formulated. Given an $r$-form $\alpha$ on a Manifold $M_2$, define the $r$-form $T^*(\alpha)$ on $M_1$ by its action on an $r$-tuple of tangent vectors $(X_1,\ldots,X_r)$ as the number $T^*(\alpha)(X_1,\ldots,X_r)=\alpha(TX_1,\ldots, TX_r)$. This defines a map on $r$-forms and is the pullback map.

See also Category