Precise definition of a linear total differential operator

Question

In the works of A. M. Vinogradov on calculus on the infinite jet space, differential equations and "diffieties", a central notion is that of a $\mathcal C$-differential operator. If $\pi:Y\rightarrow X$ is a fibered manifold, $J^\infty(\pi)$ is its infinite jet prolongation and $\mathcal F_\infty(\pi)$ is the commutative ring of smooth functions on $J^\infty(\pi)$ (defined as the direct limit of the rings $\mathcal F_r(\pi):=C^\infty(J^r(\pi))$), $P$ and $Q$ are $\mathcal F_\infty(\pi)$-modules that belong to what Vingradov calls the $\mathcal {FG}$-category, then a map $\Delta:P\rightarrow Q$ is a $\mathcal C$-differential operator if it admits restrictions to graphs of infinite jet prolongations of sections. In a coordinatized situation, these are total differential operators in the sense that they "look like" $\Delta = \sum_{|I|=0}^r\Delta^Id_I$, where the $\Delta^I$ are some matrices act by matrix multiplication on coordinatized elements of the relevant modules, while $d_i=\partial/\partial x^i+\sum y^\sigma_{iI}(\partial/\partial y^\sigma_I)$ are the coordinate total derivatives on the jet bundle.

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The above summary was inprecise and part of it is because I find Vinogradov's algebraic approach to be rather impenetrable. I'd like to give a more "differential geometry-oriented" definition of a total differential operator. Such total differential operators also appear in Anderson's "The variational bicomplex" but they are easier to characterize because they act only on evolutionary vector fields (vertical generalized vector fields) which admit a well-defined notion of prolongation.

To set up things, let's say that a (topological) vector bundle $\rho:E\rightarrow J^\infty(\pi)$ is a finite order vector bundle if there is a smooth vector bundle $\bar\rho:\bar E\rightarrow J^r(\pi)$ for some $r\in \mathbb N\cup\{-1\}$ (let's use the abuse of notation where $X=J^{-1}(\pi)$) such that $E\rightarrow J^\infty(\pi)$ is the (topological) pullback of $\bar E\rightarrow J^r(\pi)$ through the jet projection $\pi^\infty_r:J^\infty(\pi)\rightarrow J^r(\pi)$.

In practice I only care about the cases $r=-1,0$ and the $r=-1$ case is fairly trivial.

With this definition, a smooth section of $E\rightarrow J^\infty(\pi)$ is simply a "generalized section" of $\bar E\rightarrow J^r(\pi)$, i.e. a smooth map $\gamma:J^\infty(\pi)\rightarrow \bar E$ (meaning that it factors through some finite jet space) such that $\bar\rho\circ\gamma=\pi^\infty_r$.

Then if $\rho_E:E\rightarrow J^\infty(\pi)$ and $\rho_F:F\rightarrow J^\infty(\pi)$ are finite order vector bundles, a total differential operator $\Delta:\Gamma(\rho_E)\rightarrow\Gamma(\rho_F)$ from $E$ to $F$ is an $\mathbb R$-linear map that "in coordinates can be expressed only in terms of total derivatives". I'd like to find a more precise and coordinate-free version of this definition.

When both $E$ and $F$ are represented by vector bundles over $X$, this is not a problem and we can use two equivalent approaches (or at least I think so):

1. An $\mathbb R$-linear operator $\Delta:\Gamma(\rho_E)\rightarrow\Gamma(\rho_F)$ is a total differential operator if there is an ordinary linear differential operator $\bar\Delta:\Gamma(\bar\rho_E)\rightarrow \Gamma(\bar\rho_F)$ such that for any generalized section $\gamma\in \Gamma(\rho_E)$ and local section $\phi:U_{-1}\rightarrow Y$ ($U_{-1}\subseteq X$ is some open set) we have $ (\Delta(\gamma))\circ j^\infty\phi=\bar\Delta(\gamma\circ j^\infty\phi) $, which characterizes $\Delta$ completely.
2. Consider the $s$th jet prolongation $J^s(\rho_E)$ which is a vector bundle and let $\mathcal J^s(\bar\rho_E)\rightarrow J^\infty(\pi)$ denote the finite order vector bundle obtained by pulling back $J^s(\bar\rho_E)$ to $J^\infty(\pi)$. Given a generalized section $\gamma\in\Gamma(\rho_E)$, its $s$th prolongation is defined to be the section $j^s\gamma\in\Gamma(\mathcal J^s(\bar\rho_E))$ determined uniquely by $j^s\gamma\circ j^\infty\phi=j^s(\gamma\circ j^\infty\phi)$. Then an $\mathbb R$-linear map $\Delta:\Gamma(\rho_E)\rightarrow\Gamma(\rho_F)$ is a total differential operator if and only if there is a vector bundle map $\Delta_\sharp:\mathcal J^s(\bar\rho_E)\rightarrow F$ such that $\Delta(\gamma)=\Delta_\sharp\circ j^s\gamma$ (for some $s\in\mathbb N$).

But these definitions are simple because both $\rho_E$ and $\rho_F$ are induced from vector bundles over $X$.

When $\bar\rho_E:\bar E\rightarrow Y$ and $\bar\rho_F:\bar F\rightarrow Y$ are defined over say the total space $Y$, then I am not sure how to modify either definition because 1) the composite bundles $\bar E\rightarrow Y\rightarrow X$ and $\bar F\rightarrow Y\rightarrow X$ are nonlinear over $X$ so I am not sure how to define a linear differential operator between them that could be factored through the jet prolongation, 2) similar difficulties arise with the definition of the "prolongation bundle" $\mathcal J^s(\bar\rho_E)$ as $J^s(\bar\rho_E)$ is not over $X$ and $J^s(\bar E\rightarrow Y\rightarrow X)$ is not a vector bundle.

So my question is, what is the "correct" definition of a linear total differential operator between finite order vector bundles $E\rightarrow J^\infty(\pi)$ and $F\rightarrow J^\infty(\pi)$ when these are not induced from a vector bundle over $X$?

Note: Evolutionary vector fields are generalized sections of $VY\rightarrow Y$, but in this case a total differential operator can be defined as one that factors through the vertical tangent bundle $VJ^\infty(\pi)$, since there is a well-defined notion of prolongation $j^\infty\Xi$ of an evolutionary vector field. This is however specific to the properties of vector fields. The "second path" (i.e. point 2) ) aims to define a general notion of prolongation that applies to any vector bundle over $Y$.

Answer 1

I think I managed to solve this, but I have not verified this construction completely and rigorously. Still, I think it works and I am posting an answer at least for documentation and a more explicit verification later.

Compared to OP I will change notation slightly in that for this answer I am mostly dropping the notation associated with "finite order vector bundles", i.e. if $\rho:E\rightarrow J^r(\pi)$ is a vector bundle, let $\mathfrak G(\rho)$ denote the $\mathcal F_\infty(\pi)$-module of generalized sections, viz. smooth maps $\gamma:J^\infty(\pi)\rightarrow E$ that obey $\rho\circ\gamma=\pi^\infty_r$, and I will not use the pullback bundle $(\pi^\infty_r)^\ast E$ unless absolutely necessary.

Furthermore, another construction is useful. Fix an integer $r_0\in\mathbb N$ and suppose that for each $r\ge r_0$ we are given a vector bundle $\rho_r:E_r\rightarrow J^r(\pi)$, and for each $r\ge s$ we also have a vector bundle morphism $\eta^r_s:E_r\rightarrow E_s$ over $\pi^r_s$. Although not necessary to assume this, for applications it is safe to assume that $\eta^r_s$ is a fibration. Then the system $(E_r,\eta^r_s)$ of vector bundles is an inverse system and its projective limit $(E_\infty,\eta^\infty_r)$ is a vector bundle over $J^\infty(\pi)$. Such vector bundles will be called projective vector bundles and the "formal" smooth structure on $E_\infty$ is analogous to the one on $J^\infty(\pi)$. Then a smooth section of $E_\infty\rightarrow J^\infty(\pi)$ is equivalent to a sequence $(\gamma_{r_0},\gamma_{r_0+1},\dots)$ of generalized sections $\gamma_r\in\mathfrak G(\rho_r)$ such that for each $r\ge s$ we have $\gamma_s=\eta^r_s\circ\gamma_r$. This construction is completely analogous to that of the tangent bundle $TJ^\infty(\pi)$.

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Let now $\rho:E\rightarrow J^s(\pi)$ be a vector bundle and $\rho_X:=\pi^s\circ\rho:E\rightarrow J^s(\pi)\rightarrow X$ be the composite bundle.

It can be seen that the $r$th prolongation $(\rho_X)^r:J^r(\rho_X)\rightarrow X$ factors through the fibered manifold $j^r\rho:J^r(\rho_X)\rightarrow J^r(\pi^s)$, where $J^r(\pi^s)=J^rJ^s(\pi)$ is a non-holonomic jet bundle, moreover, this is a vector bundle. Let $i_{r,s}:J^{r+s}(\pi)\rightarrow J^r(\pi^s)$ be the natural embedding of the $r+s$th holonomic jet space into the $(r,s)$ non-holonomic jet space, and let $$ \mathrm{pr}^r(\rho):\mathrm{Pr}^r(\rho)\rightarrow J^{r+s}(\pi),\quad\mathrm{Pr}^r(\rho)=(i_{r,s}^\ast J^r(\rho_X)) $$ be the pullback vector bundle to the holonomic jet space.

Given a section $\gamma\in\Gamma(\rho):J^s(\pi)\rightarrow E$, there is a unique section $\mathrm{pr}^r(\gamma):J^{r+s}(\pi)\rightarrow\mathrm{Pr}^r(\rho)$ which is given by $$ [\mathrm{pr}^r(\gamma)](j^{r+s}_x\phi)=j^r_x(\gamma\circ j^s\phi), $$ and we may call $\mathrm{pr}^r(\gamma)$ the $r$th prolongation of $\gamma$.

Prolongation is also well-defined for generalized sections. If $\gamma\in\mathfrak G(\rho)$ is a generalized section, then $\mathrm{pr}^r(\gamma)\in\mathfrak G(\mathrm{pr}^r(\rho))$ is the unique generalized section of $\mathrm{Pr}^r(\rho)\rightarrow J^{r+s}(\pi)$ that satisfies $$ [\mathrm{pr}^r(\gamma)]\circ j^\infty\phi=j^r(\gamma\circ j^\infty\phi) $$ for any local section $\phi\in\Gamma_\text{loc}(\pi)$.

For each $r\ge r^\prime$, $\mathrm{Pr}^r(\rho)$ projects onto $\mathrm{Pr}^{r^\prime}(\rho)$, so these "prolongation bundles" determine a projective vector bundle $\mathrm{Pr}(\rho)\rightarrow J^\infty(\pi)$, whose sections are the infinite prolongations $\mathrm{pr}(\gamma)=(\gamma,\mathrm{pr}^1(\gamma),\dots)$ of generalized sections.

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Finally, let $\rho_E:E\rightarrow J^r(\pi)$ and $\rho_F:F\rightarrow J^s(\pi)$ be a pair of vector bundles. A total differential operator $\Delta:\mathfrak G(\rho_E)\rightarrow\mathfrak G(\rho_F)$ from (generalized sections of) $E$ to $F$ is an $\mathbb R$-linear operator such that there exists a smooth vector bundle morphism $\Delta_\sharp:\mathrm{Pr}(\rho_E)\rightarrow F$ (over $\pi^\infty_s$; smooth meaning it factors through some finite prolongation bundle) such that for any $\gamma\in\mathfrak G(\rho_E)$ we have $$ \Delta(\gamma)=\Delta_\sharp\circ\mathrm{pr}(\gamma). $$

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Note that if $E\rightarrow J^r(\pi)=VY\rightarrow Y$, then this notion of prolongation reproduces the prolongation of evolutionary vector fields, but it does not reproduce the prolongation of non-vertical (generalized) vector fields over $Y$.