# Pullback of a Vector-valued form treated as Differential Operator

Suppose we are given a vector bundle $$E\to M$$ and a $$E$$-valued $$p$$-form, $$\omega\in\Gamma(E\otimes \Lambda^pT^*M)$$ For any smooth $$f:\Sigma\to M$$ then we have the pullback $$p$$-form, $$f^*\omega\in\Gamma(f^*E\otimes\Lambda^p T^*\Sigma)$$ Now we may consider an operator, $$D: f\mapsto f^*\omega$$. This $$D$$ is a first order ''differential operator''. I can see this if I work with a local trivialization of the bundle $$E$$.

My question is how to see this $$D$$ as a global operator? More precisely, I wish to write $$D$$ as an operator between section spaces, \begin{align*}D:C^\infty(\Sigma,M)&\to\Gamma(X)\\ f&\mapsto f^*\omega\end{align*} I am unable to figure out the space $$X$$ above.

Any help regarding this appreciated. Thank you!

• You already wrote $f^*\omega \in \Gamma(f^*E\otimes\bigwedge^pT^*\Sigma)$, so why isn't $X = f^*E\otimes\bigwedge^pT^*\Sigma$? Feb 28, 2020 at 21:41
• @MichaelAlbanese Because $X$ depends on the function $f$, no? Feb 29, 2020 at 6:19
• Ah, I see. In that case, $D$ is not a differential operator right (at least according to the definition I have in mind)? If you disagree, would you mind including your definition of a differential operator? Feb 29, 2020 at 12:25
• @MichaelAlbanese Yes, I agree that $D$ cannot be a lienar operator between sections of vector bundles. But it still has a local description which gives rise to a first order PDE. In fact, if I had that $E$ is a trivial vector bundle of rank $k$, then $D:C^\infty(\Sigma,M)\to \Gamma(\Lambda^p T^*M \otimes \mathbb{R}^k)$ is indeed a differential operator of order $1$. Feb 29, 2020 at 14:16

In order to formalize the differential operator in a global setup, consider the space $$\mathcal{B}=C^\infty(\Sigma, M)$$. For $$f:\Sigma\to M$$, denote the space $$\mathcal{E}_f=\Gamma \big(f^*E\otimes \Lambda^pT^*\Sigma\big)$$. Then we can think of a bundle $$\mathcal{E}\to\mathcal{B}$$. This is an infinite dimensional vector bundle, where the spaces have Frechet topology. The operator $$D:f\mapsto f^*\omega$$ can then be realized as a section of this bundle. This seems like the correct way to study the operator.
As an aside, one can get the linearization operator associated to $$D$$ at $$f$$, by considering some arbitrary connection $$\nabla$$ on $$E$$. One then gets the operator, \begin{align*}L_f : \Gamma f^*TM &\to \Gamma\hom(\Lambda^p TV, f^*E) \\ \xi &\mapsto \mathfrak{L}_\xi\omega\end{align*} Here $$\mathfrak{L}$$ is the Lie derivative of $$\omega$$ along $$\xi$$ and is given by the Cartan formula, $$\mathfrak{L}_\xi\omega = \iota_\xi d_\nabla\omega + d_\nabla\iota_\xi\omega$$. The details for Lie derivative by a vector field along a map, may found in Michor's book.