Editing Differential form (section)

=== Integration along fibers ===
{{main|Integration along fibers}}

[[Fubini's theorem]] states that the integral over a set that is a product may be computed as an iterated integral over the two factors in the product.  This suggests that the integral of a differential form over a product ought to be computable as an iterated integral as well.  The geometric flexibility of differential forms ensures that this is possible not just for products, but in more general situations as well.  Under some hypotheses, it is possible to integrate along the fibers of a smooth map, and the analog of Fubini's theorem is the case where this map is the projection from a product to one of its factors.

Because integrating a differential form over a submanifold requires fixing an orientation, a prerequisite to integration along fibers is the existence of a well-defined orientation on those fibers.  Let {{math|''M''}} and {{math|''N''}} be two orientable manifolds of pure dimensions {{math|''m''}} and {{math|''n''}}, respectively.  Suppose that {{math|''f'' : ''M'' → ''N''}} is a surjective submersion.  This implies that each fiber {{math|''f''{{i sup|&minus;1}}(''y'')}} is {{math|(''m'' &minus; ''n'')}}-dimensional and that, around each point of {{math|''M''}}, there is a chart on which {{math|''f''}} looks like the projection from a product onto one of its factors.  Fix {{math|''x'' ∈ ''M''}} and set {{math|1=''y'' = ''f''(''x'')}}.  Suppose that
<math display="block">\begin{align}
\omega_x &\in {\textstyle\bigwedge}^m T_x^*M, \\[2pt]
\eta_y &\in {\textstyle\bigwedge}^n T_y^*N,
\end{align}</math>
and that {{math|''η''<sub>''y''</sub>}} does not vanish.  Following {{Harv|Dieudonné|1972}}, there is a unique
<math display="block">\sigma_x \in {\textstyle\bigwedge}^{m-n} T_x^*(f^{-1}(y))</math>
which may be thought of as the fibral part of {{math|''ω''<sub>''x''</sub>}} with respect to {{math|''η''<sub>''y''</sub>}}.  More precisely, define {{math|''j'' : ''f''{{i sup|&minus;1}}(''y'') → ''M''}} to be the inclusion.  Then {{math|''σ''<sub>''x''</sub>}} is defined by the property that
<math display="block">\omega_x = (f^*\eta_y)_x \wedge \sigma'_x \in {\textstyle\bigwedge}^m T_x^*M,</math>
where
<math display="block">\sigma'_x \in {\textstyle\bigwedge}^{m-n} T_x^*M</math>
is any {{math|(''m'' &minus; ''n'')}}-covector for which
<math display="block">\sigma_x = j^*\sigma'_x.</math>
The form {{math|''σ''<sub>''x''</sub>}} may also be notated {{math|''ω''<sub>''x''</sub> / ''η''<sub>''y''</sub>}}.

Moreover, for fixed {{math|''y''}}, {{math|''σ''<sub>''x''</sub>}} varies smoothly with respect to {{math|''x''}}.  That is, suppose that
<math display="block">\omega \colon f^{-1}(y) \to T^*M</math>
is a smooth section of the projection map; we say that {{math|''ω''}} is a smooth differential {{math|''m''}}-form on {{math|''M''}} along {{math|''f''{{i sup|&minus;1}}(''y'')}}.  Then there is a smooth differential {{math|(''m'' &minus; ''n'')}}-form {{math|''σ''}} on {{math|''f''{{i sup|&minus;1}}(''y'')}} such that, at each {{math|''x'' ∈ ''f''{{i sup|&minus;1}}(''y'')}},
<math display="block">\sigma_x = \omega_x / \eta_y.</math>
This form is denoted {{math|''ω'' / ''η''<sub>''y''</sub>}}.  The same construction works if {{math|''ω''}} is an {{math|''m''}}-form in a neighborhood of the fiber, and the same notation is used.  A consequence is that each fiber {{math|''f''{{i sup|&minus;1}}(''y'')}} is orientable.  In particular, a choice of orientation forms on {{math|''M''}} and {{math|''N''}} defines an orientation of every fiber of {{math|''f''}}.

The analog of Fubini's theorem is as follows.  As before, {{math|''M''}} and {{math|''N''}} are two orientable manifolds of pure dimensions {{math|''m''}} and {{math|''n''}}, and {{math|''f'' : ''M'' → ''N''}} is a surjective submersion.  Fix orientations of {{math|''M''}} and {{math|''N''}}, and give each fiber of {{math|''f''}} the induced orientation.  Let {{math|''ω''}} be an {{math|''m''}}-form on {{math|''M''}}, and let {{math|''η''}} be an {{math|''n''}}-form on {{math|''N''}} that is almost everywhere positive with respect to the orientation of {{math|''N''}}.  Then, for almost every {{math|''y'' ∈ ''N''}}, the form {{math|''ω'' / ''η''<sub>''y''</sub>}} is a well-defined integrable {{math|''m'' &minus; ''n''}} form on {{math|''f''{{i sup|&minus;1}}(''y'')}}.  Moreover, there is an integrable {{math|''n''}}-form on {{math|''N''}} defined by
<math display="block">y \mapsto \bigg(\int_{f^{-1}(y)} \omega / \eta_y\bigg)\,\eta_y.</math>
Denote this form by
<math display="block">\bigg(\int_{f^{-1}(y)} \omega / \eta\bigg)\,\eta.</math>
Then {{Harv|Dieudonné|1972}} proves the generalized Fubini formula
<math display="block">\int_M \omega = \int_N \bigg(\int_{f^{-1}(y)} \omega / \eta\bigg)\,\eta.</math>

It is also possible to integrate forms of other degrees along the fibers of a submersion.  Assume the same hypotheses as before, and let {{math|''α''}} be a compactly supported {{math|(''m'' &minus; ''n'' + ''k'')}}-form on {{math|''M''}}.  Then there is a {{math|''k''}}-form {{math|''γ''}} on {{math|''N''}} which is the result of integrating {{math|''α''}} along the fibers of {{math|''f''}}.  The form {{math|''α''}} is defined by specifying, at each {{math|''y'' ∈ ''N''}}, how {{math|''γ''}} pairs with each {{math|''k''}}-vector {{math|'''v'''}} at {{math|''y''}}, and the value of that pairing is an integral over {{math|''f''{{i sup|&minus;1}}(''y'')}} that depends only on {{math|''α''}}, {{math|'''v'''}}, and the orientations of {{math|''M''}} and {{math|''N''}}.  More precisely, at each {{math|''y'' ∈ ''N''}}, there is an isomorphism
<math display="block">{\textstyle\bigwedge}^k T_yN \to {\textstyle\bigwedge}^{n-k} T_y^*N</math>
defined by the interior product
<math display="block">\mathbf{v} \mapsto \mathbf{v}\,\lrcorner\,\zeta_y,</math>
for any choice of volume form {{math|''ζ''}} in the orientation of {{math|''N''}}. If {{math|''x'' ∈ ''f''{{i sup|&minus;1}}(''y'')}}, then a {{math|''k''}}-vector {{math|'''v'''}} at {{math|''y''}} determines an {{math|(''n'' &minus; ''k'')}}-covector at {{math|''x''}} by pullback:
<math display="block">f^*(\mathbf{v}\,\lrcorner\,\zeta_y) \in {\textstyle\bigwedge}^{n-k} T_x^*M.</math>
Each of these covectors has an exterior product against {{math|''α''}}, so there is an {{math|(''m'' &minus; ''n'')}}-form {{math|''β''<sub>'''v'''</sub>}} on {{math|''M''}} along {{math|''f''{{i sup|&minus;1}}(''y'')}} defined by
<math display="block">(\beta_{\mathbf{v}})_x = \left(\alpha_x \wedge f^*(\mathbf{v}\,\lrcorner\,\zeta_y)\right) \big/ \zeta_y \in {\textstyle\bigwedge}^{m-n} T_x^*M.</math>
This form depends on the orientation of {{math|''N''}} but not the choice of {{math|''ζ''}}.  Then the {{math|''k''}}-form {{math|''γ''}} is uniquely defined by the property
<math display="block">\langle\gamma_y, \mathbf{v}\rangle = \int_{f^{-1}(y)} \beta_{\mathbf{v}},</math>
and {{math|''γ''}} is smooth {{Harv|Dieudonné|1972}}.  This form also denoted {{math|''α''<sup>♭</sup>}} and called the '''integral of {{math|''α''}} along the fibers of {{math|''f''}}'''.  Integration along fibers is important for the construction of Gysin maps in de Rham cohomology.

Integration along fibers satisfies the '''projection formula''' {{Harv|Dieudonné|1972}}.  If {{math|''λ''}} is any {{math|''ℓ''}}-form on {{math|''N''}}, then
<math display="block">\alpha^\flat \wedge \lambda = (\alpha \wedge f^*\lambda)^\flat.</math>