Pullback (differential Geometry) - Pullback of Cotangent Vectors and 1-forms

Pullback of Cotangent Vectors and 1-forms

Let φ : M → N be a smooth map between smooth manifolds. Then the differential of φ, φ_* = dφ (or Dφ), is a vector bundle morphism (over M) from the tangent bundle TM of M to the pullback bundle φ*TN. The transpose of φ_* is therefore a bundle map from φ*T*N to T*M, the cotangent bundle of M.

Now suppose that α is a section of T*N (a 1-form on N), and precompose α with φ to obtain a pullback section of φ*T*N. Applying the above bundle map (pointwise) to this section yields the pullback of α by φ, which is the 1-form φ*α on M defined by

for x in M and X in T_xM.