<Wiki>
The notion of pullback in mathematics is a fundamental one. It refers to two different, but related processes: precomposition and fiber-product.
Precomposition with a function probably provides the most elementary notion of pullback: in simple terms, a function f of a variable y, where y itself is a function of another variable x, may be written as a function of x. This is the pullback of f by the function y(x).
Let φ:M→ N be a smooth map between (smooth) manifolds M and N, and suppose f:N→R is a smooth function on N. Then the pullback of f by φ is the smooth function φ*f on M defined by (φ*f)(x) = f(φ(x)). Similarly, if f is a smooth function on an open set U in N, then the same formula defines a smooth function on the open set φ-1(U) in M. (In the language of sheaves, pullback defines a morphism from the sheaf of smooth functions on N to the direct image by φ of the sheaf of smooth functions on M.)
More generally, if f:N→A is a smooth map from N to any other manifold A, then φ*f(x)=f(φ(x)) is a smooth map from M to A.
<Frankel> p.57
The presence of the Poincare’ 1-form field on T*M and the capability of pulling back 1-form fields under mappings endow T*M with a powerful tool that is not available on TM.
