Differential Form - Pullback

Pullback

One of the main reasons the cotangent bundle rather than the tangent bundle is used in the construction of the exterior complex is that differential forms are capable of being pulled back by smooth maps, while vector fields cannot be pushed forward by smooth maps unless the map is, say, a diffeomorphism. The existence of pullback homomorphisms in de Rham cohomology depends on the pullback of differential forms.

Differential forms can be moved from one manifold to another using a smooth map. If f : M → N is smooth and ω is a smooth k-form on N, then there is a differential form f*ω on M, called the pullback of ω, which captures the behavior of ω as seen relative to f.

To define the pullback, recall that the differential of f is a map f_* : TM → TN. Fix a differential k-form ω on N. For a point p of M and tangent vectors v₁, ..., v_k to M at p, the pullback of ω is defined by the formula

More abstractly, if ω is viewed as a section of the cotangent bundle T*N of N, then f*ω is the section of T*M defined as the composite map

Pullback respects all of the basic operations on forms:

The pullback of a form can also be written in coordinates. Assume that x₁, ..., x_m are coordinates on M, that y₁, ..., y_n are coordinates on N, and that these coordinate systems are related by the formulas y_i = f_i(x₁, ..., x_m) for all i. Then, locally on N, ω can be written as

where, for each choice of i₁, ..., i_k, is a real-valued function of y₁, ..., y_n. Using the linearity of pullback and its compatibility with wedge product, the pullback of ω has the formula

Each exterior derivative df_i can be expanded in terms of dx₁, ..., dx_m. The resulting k-form can be written using Jacobian matrices: