**A geometric way of looking at differential equations**

In the literature for the h-principle (for example Gromov's *Partial differential relations* or Eliashberg and Mishachev's *Introduction to the h-principle*), we often see the following (all objects smooth):

Give a fibre bundle $\pi:F\to M$ over some manifold $M$, denote by $F^{(r)}$ the associated $r$-jet bundle. A

partial differential relation$\mathcal{R}$ is a subset of $F^{(r)}$.$\mathcal{R}$ is said to be a

partial differential equationif it is a submanifold of $F^{(r)}$ with codimension $\geq 1$.A

solution$\Phi$ to the relation $\mathcal{R}$ is aholonomic(in the sense that $\Phi = j^r\phi$ for some section $\phi$ of $F$) section of $F^{(r)}$ that lies in $\mathcal{R}$.

This description is very powerful in the context of the topologically motivated techniques of the h-principle. And for partial differential inequalities where $\mathcal{R}$ is an open submanifold, this allows the convenient setting for the Holonomic Approximation Theorem.

**A geometric way of looking at differential operators**

Here I recall the famous theorem of Peetre.

Let $E\to M$ and $F\to M$ be two vector bundles. Let $D$ be a linear operator mapping sections of $E$ to sections of $F$. Suppose $D$ is support-non-increasing, then $D$ is (locally) a linear differential operator.

where

A

linear differential operator$D:\Gamma E\to\Gamma F$ is a composition $D := i\circ j$ where $j: E\to J^RE$ is the $R$-jet operator, and $i: J^RE \to F$ is a linear map of vector bundles.

Most of the time in applications, $F$ can be taken to be the same bundle as $E$. This way of phrasing things is also convenient for analysis. For example, we can easily define the **principal symbol** of a linear differential operator in the following way.

Let $D_1$ and $D_2$ be two linear differential operators of order $r$ (that is, $r$ is the smallest natural number such that if $j^r\phi = j^r\psi$, then $D\phi = D\psi$). Let $\pi^r_{r-1}: J^rE \to J^{r-1}E$ the natural projection. We say that $D_1$ and $D_2$ have the same principal part if their corresponding $i_1 - i_2|_{\ker \pi^{r}_{r-1}} \equiv 0$. (In words, their difference is a l.d.o. of lower order. This defines an equivalence relation on linear differential operators of order $r$. Each equivalence class defines a unique linear map of vector bundles $P: \ker \pi^{r}_{r-1} \to F$.

Now, it is known (sec 12.10 in Kolar, Michor, Slovak's *Natural operations...*) that $\ker \pi^{r}_{r-1}$ is canonically isomorphic to $E\otimes S^r(T^*M)$, where $S^r$ denotes the $r$-fold symmetric tensor product of a space with itself. So we have naturally an interpretation of $P$ as a section of $F\otimes E^* \otimes S^r(TM)$. In the case where $E$ and $F$ are just, say, the trivial $\mathbb{R}$ bundle over $M$, we recover the usual description of the principle symbol of a pseudo-differential operator being (fibre-wise) a homogeneous polynomial of the cotangent space.

**Question**

Given the above, another way of looking at partial differential equations is *perhaps* the following.

Let $\pi_X: X\to M$ and $\pi_Y: Y\to M$ be fibre bundles. A **system of partial differential operators** of order $r$ is defined to be a map $H: J^rX \to Y$ that commutes with projection $\pi_X \circ \pi^r_0 = \pi_Y \circ H$. And a **system of partial differential equations** is just the statement $H(j^r\phi) = \psi$, where $\phi\in\Gamma X$ and $\psi \in\Gamma Y$. Observe that by considering $H^{-1}(\psi) \subset J^rX$, we clearly have a partial differential relation the sense defined in the first section. If we require that $H$ is a submersion, then $H^{-1}(\psi)$ is also a partial differential equation in the sense defined before.

On the other hand, if $\mathcal{R}$ is an embedded submanifold of $J^rX$, at least locally $\mathcal{R}$ can be written as the pre-image of a point of a submersion; though there may be problems making this a global description. So perhaps my definition is in fact more restrictive than the one given in the first part of this discussion.

**My question is**: is this last "definition" discussed anywhere in the literature? Perhaps with its pros and cons versus the partial differential relations definition given in the first part of the question? I am especially interested in references taking a more PDE (as opposed to differential geometry) point of view, but any reference at all would be welcome.

*Note:* For the `reference-request`

part of this question, I would also appreciate pointers to whom I can ask/e-mail on these matters.

at the momentthis will be formal set-up that will end up being ¨tossed away¨; so at the moment I am just trying to cover my behind by making sure I am not saying anything irresponsible.`:-)`

(Though this may become genuinely the focus later.) As to why I need this formal set-up: I want to get a good notion of a ¨quasilinear¨ partial differential equationindependentof the choice of connection, and to define its linearisation and its linearised principal symbol. $\endgroup$ – Willie Wong Sep 28 '11 at 15:23