Editing Differential form (section)

== Pullback ==
{{see also|Pullback (differential geometry)}}

Suppose that {{math|''f'' : ''M'' → ''N''}} is smooth. The differential of {{math|''f''}} is a smooth map {{math|''df'' : ''TM'' → ''TN''}} between the tangent bundles of {{math|''M''}} and {{math|''N''}}. This map is also denoted {{math|''f''<sub>∗</sub>}} and called the '''pushforward'''. For any point {{math|''p'' ∈ ''M''}} and any tangent vector {{math|''v'' &isin; ''T''<sub>''p''</sub>''M''}}, there is a well-defined pushforward vector {{math|''f''<sub>∗</sub>(''v'')}} in {{math|''T''<sub>''f''(''p'')</sub>''N''}}. However, the same is not true of a vector field.  If {{math|''f''}} is not injective, say because {{math|''q'' ∈ ''N''}} has two or more preimages, then the vector field may determine two or more distinct vectors in {{math|''T''<sub>''q''</sub>''N''}}. If {{math|''f''}} is not surjective, then there will be a point {{math|''q'' &isin; ''N''}} at which {{math|''f''<sub>∗</sub>}} does not determine any tangent vector at all. Since a vector field on {{math|''N''}} determines, by definition, a unique tangent vector at every point of {{math|''N''}}, the pushforward of a vector field does not always exist.

By contrast, it is always possible to pull back a differential form.  A differential form on {{math|''N''}} may be viewed as a linear functional on each tangent space.  Precomposing this functional with the differential {{math|''df'' : ''TM'' → ''TN''}} defines a linear functional on each tangent space of {{math|''M''}} and therefore a differential form on {{math|''M''}}.  The existence of pullbacks is one of the key features of the theory of differential forms.  It leads to the existence of pullback maps in other situations, such as pullback homomorphisms in de Rham cohomology.

Formally, let {{math|''f'' : ''M'' → ''N''}} be smooth, and let {{math|''ω''}} be a smooth {{math|''k''}}-form on {{math|''N''}}.  Then there is a differential form {{math|''f''{{i sup|∗}}''ω''}} on {{math|''M''}}, called the '''pullback''' of {{math|''ω''}}, which captures the behavior of {{math|''ω''}} as seen relative to {{math|''f''}}.  To define the pullback, fix a point {{math|''p''}} of {{math|''M''}} and tangent vectors {{math|''v''<sub>1</sub>}}, ..., {{math|''v''<sub>''k''</sub>}} to {{math|''M''}} at {{math|''p''}}.  The pullback of {{math|''ω''}} is defined by the formula
<math display="block">(f^*\omega)_p(v_1, \ldots, v_k) = \omega_{f(p)}(f_*v_1, \ldots, f_*v_k).</math>

There are several more abstract ways to view this definition.  If {{math|''ω''}} is a {{math|1}}-form on {{math|''N''}}, then it may be viewed as a section of the cotangent bundle {{math|''T''{{i sup|∗}}''N''}} of {{math|''N''}}.  Using {{i sup|∗}} to denote a dual map, the dual to the differential of {{math|''f''}} is {{math|(''df''){{i sup|∗}} : ''T''{{i sup|∗}}''N'' → ''T''{{i sup|∗}}''M''}}.  The pullback of {{math|''ω''}} may be defined to be the composite
<math display="block">M\ \stackrel{f}{\to}\ N\ \stackrel{\omega}{\to}\ T^*N\ \stackrel{(df)^*}{\longrightarrow}\ T^*M.</math>
This is a section of the cotangent bundle of {{math|''M''}} and hence a differential {{math|1}}-form on {{math|''M''}}.  In full generality, let <math display="inline">\bigwedge^k (df)^*</math> denote the {{math|''k''}}th exterior power of the dual map to the differential.  Then the pullback of a {{math|''k''}}-form {{math|''ω''}} is the composite
<math display="block">M\ \stackrel{f}{\to}\ N\ \stackrel{\omega}{\to}\ {\textstyle\bigwedge}^k T^*N\ \stackrel{{\bigwedge}^k (df)^*}{\longrightarrow}\ {\textstyle\bigwedge}^k T^*M.</math>

Another abstract way to view the pullback comes from viewing a {{math|''k''}}-form {{math|''ω''}} as a linear functional on tangent spaces.  From this point of view, {{math|''ω''}} is a morphism of [[vector bundle]]s
<math display="block">{\textstyle\bigwedge}^k TN\ \stackrel{\omega}{\to}\ N \times \mathbf{R},</math>
where {{math|''N'' × '''R'''}} is the trivial rank one bundle on {{math|''N''}}.  The composite map
<math display="block">{\textstyle\bigwedge}^k TM\ \stackrel{{\bigwedge}^k df}{\longrightarrow}\ {\textstyle\bigwedge}^k TN\ \stackrel{\omega}{\to}\ N \times \mathbf{R}</math>
defines a linear functional on each tangent space of {{math|''M''}}, and therefore it factors through the trivial bundle {{math|''M'' × '''R'''}}.  The vector bundle morphism <math display="inline">{\textstyle\bigwedge}^k TM \to M \times \mathbf{R}</math> defined in this way is {{math|''f''{{i sup|∗}}''ω''}}.

Pullback respects all of the basic operations on forms.  If {{math|''ω''}} and {{math|''η''}} are forms and {{math|''c''}} is a real number, then
<math display="block">\begin{align}
             f^*(c\omega) &= c(f^*\omega), \\
       f^*(\omega + \eta) &= f^*\omega + f^*\eta, \\
  f^*(\omega \wedge \eta) &= f^*\omega \wedge f^*\eta, \\
             f^*(d\omega) &= d(f^*\omega).
\end{align}</math>

The pullback of a form can also be written in coordinates. Assume that {{math|''x''<sup>1</sup>}}, ..., {{math|''x''<sup>''m''</sup>}} are coordinates on {{math|''M''}}, that {{math|''y''<sup>1</sup>}}, ..., {{math|''y''<sup>''n''</sup>}} are coordinates on {{math|''N''}}, and that these coordinate systems are related by the formulas {{math|1=''y''<sup>''i''</sup> = ''f''<sub>''i''</sub>(''x''<sup>1</sup>, ..., ''x''<sup>''m''</sup>)}} for all {{math|''i''}}. Locally on {{math|''N''}}, {{math|''ω''}} can be written as
<math display="block">\omega = \sum_{i_1 < \cdots < i_k} \omega_{i_1\cdots i_k} \, dy^{i_1} \wedge \cdots \wedge dy^{i_k},</math>
where, for each choice of {{math|''i''<sub>1</sub>}}, ..., {{math|''i''<sub>''k''</sub>}}, {{math|''ω''<sub>''i''<sub>1</sub>⋅⋅⋅''i''<sub>''k''</sub></sub>}} is a real-valued function of {{math|''y''<sup>1</sup>}}, ..., {{math|''y''<sup>''n''</sup>}}. Using the linearity of pullback and its compatibility with exterior product, the pullback of {{math|''ω''}} has the formula
<math display="block">f^*\omega = \sum_{i_1 < \cdots < i_k} (\omega_{i_1\cdots i_k}\circ f) \, df_{i_1} \wedge \cdots \wedge df_{i_k}.</math>

Each exterior derivative {{math|''df''<sub>''i''</sub>}} can be expanded in terms of {{math|''dx''<sup>1</sup>}}, ..., {{math|''dx''<sup>''m''</sup>}}. The resulting {{math|''k''}}-form can be written using [[Jacobian matrix and determinant|Jacobian]] matrices:
<math display="block"> f^*\omega = \sum_{i_1 < \cdots < i_k} \sum_{j_1 < \cdots < j_k} (\omega_{i_1\cdots i_k}\circ f)\frac{\partial(f_{i_1}, \ldots, f_{i_k})}{\partial(x^{j_1}, \ldots, x^{j_k})} \, dx^{j_1} \wedge \cdots \wedge dx^{j_k}.</math>

Here, <math display="inline>\frac{\partial(f_{i_1}, \ldots, f_{i_k})}{\partial(x^{j_1}, \ldots, x^{j_k})}</math> denotes the determinant of the matrix whose entries are <math display="inline">\frac{\partial f_{i_m}}{\partial x^{j_n}}</math>, <math>1\leq m,n\leq k</math>.