I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold.
Let $\pi: E \to M$ be a smooth locally trivial fibre bundle. In Gromovs words a partial differential relation of order $k$ is a subset of the $k$th jet bundle $J^k(E)$. Usually one defines a partial differential equation of order $k$ to be a closed submanifold of the $k$th jet bundle $J^k(E)$. This last definition brings in some kind of regularity condition. I do not really understand why one wants/needs this regularity.
What is the advantage of asking a PDE to be a submanifold in lieu of beeing merely a subset of $J^k(E)$?
Let me explain a bit further: (First, remark that this is somehow a follow up for my earlier question: Why is a differential equation a submanifold of a jet bundle?)
Often a partial differntial relation comes from a differential operator. Let $\rho: H \to M$ be another fibred manifold. A differential operator of order $k$ is a map $D_f: \Gamma_{loc}(\pi) \to \Gamma_{loc}(\rho)$ between local sections, for which there exists a bundle morphism $f: J^k(E) \to H$ such that for every local section $\phi \in \Gamma_{loc}(\pi)$ the equality $D_f (\phi) (p) = f(j^k_p \phi)$ holds for all $p$ in the domain of $\phi$. Now the preimage of a given section $\eta$ of $\rho$ under $f$ is clearly a partial differential relation, but for it to be a partial differntial equation we need to ask $df$ to have constant rank.
This previous paragraph is a bit opposed to my naiv interpretation of a PDE in $R^n$ to be an arbitrary equation in the indpendent variable, the funtion itself and it's partial derivatives. Just in the sense of wikipedia: A partial differential equation looks like $F(x_1, \dots, x_n, u(x), \frac{\partial u}{\partial x_1}, \dots ) = 0$ for an arbitrary function $F$ (with image, say, in $R$). Now, if we are looking for functions $u: R^n \to R^m$ we can set $E = R^n \times R^m$ with $\pi$ beeing the projection on the first factor and $H = R^n \times R$. $F$ is a bundle map, that implicitly also defines a differential operator $D_F$. So in the stricter sense of beeing a submanifold, $F$ defines only a partial differential equation, if it hat constant rank, otherwise it is "merely" a partial differential relation.
Edit: Example: Why is such a thing as $\frac{\partial u}{\partial x} \frac{\partial u}{\partial y} = 0$ not a pde (in the narrower sense)?
In this case $(E, M, \pi) = (R^2 \times R, R^2, pr_2)$ with coordinates $(x_1, x_2, y)$. The first jet bundle $J^1(E)$ has coordinates $(x_1, x_2, y, p_1, p_2)$. The partial differential relation $S = \lbrace (x_1, x_2, y, p_1, p_2) \, | \, p_1 p_2 = 0 \rbrace$ describes the equation I started with and is clearly not a smooth submanifold.
Why is this gap between the naiv real analysis point of view and the somewhat more elaboratet differential geometric point of view?
Last I'm curious if there is a nice way to see, that a specific relation is in deed a partial differential equation. For example the equality $\Phi^* g = g$ specifying local diffeomorphisms that are isometries with respect to a pseudo-Riemannian metric $g$ is often said to be a partial differential equation, without giving an argument. And in parts I agree: this really feels like a partial differential equation. But to show that it defines in deed a submanifold needs some works, as can be seen in my earlier question Why is $\phi^* g = g$ a PDE for a pseudo-Riemannian metric $g$ on a manifold?, which got a very nice answer. Other examples that are often said to be defined by partial differential equations are locally defined affine maps with respect to an arbitrary connection $\nabla$ in $TM$ or locally defined diffeomorphisms that preserve a given tensor field $T$ or even those affine diffeomorphisms that in addtion preserve a given tensor field. Maybe it would be helpful to see an argument in coordinates in one of this cases, to see how this intuitivly obvious statements can be made rigorous.
Edit: To start with the "coorinate argument" I asked for in the last paragraph, set $E=M \times M$ and $\pi$ the projection to the first factor and we can restrict ourselfes to the open submanifold of $J^1(E)$, consisting of 1-jets of local diffeomorphisms. Note that for $(x,U)$ a chart arount $p$ and $(y, V)$ a chart arount $f(p)$, a chart arount $j^1_p f$ is given by $\phi_{xy}: j^1_q g \to (x(q), y(g(q)), d(y \circ g \circ x^{-1})_{x(q)}) \in x(U) \times y(V) \times GL(\Bbb R^n)$. Remark, that for $\phi_{xy}(j^1_q g) = (\xi, \eta, A)$ and $\Phi (\mu) := A(\mu) + \eta - A(\xi)$ the function $y^{-1} \circ \Phi \circ x$ (restricted to a suitable subset) is a representative of the equivalence class $j^1_q g$.
Now we can pullback the connection $\nabla$ and/or the tensorfield $T$ to give connections/tensor fields $\nabla^x, \nabla^y, T^x, T^y$ on $x(U)$, $y(v)$ via $x^{-1}, y^{-1}$ respectively. The property to preserve the connection/tensor field spells out in coordinates as \begin{align*} F(\xi, \eta, A) &= (\Phi^* T^y)_{\xi} - T^x_{\xi} = 0 \\ G(\xi, \eta, A) &= (\Phi^* \nabla^y)_{\xi} - \nabla^x_{\xi} = 0 \end{align*}
(The first term of $F$ resp. $G$ depends only on $\eta$ and $A$, whereas the second term depends only on $\xi$.) This can be translatet into equations for the components of $T^x$ and $T^y$ resp. the Christoffel symbols of $\nabla^x, \nabla^y$ with respect to the standard coordniate frame of $\Bbb R^n$.
So far so good. But now I need to proove, that the rank of the differentials of $F, G, (F,G)$ respectively is constant for solutions and moreover does neither depend on the choice of chart, nor of the domain/image of the local solution $g$. Any hints, how to do this in any of this cases?
Edit 2: As Hubert Goldschmidt points out in his 1967 article Integrability criteria for systems of nonlinear partial differential equations, it suffices that the bundle morphism defining a differential operator has locally constant rank. That means in the above discussion, it suffices to show, that $F$ etc. have constant rank for all pairs of charts $((x,U),(y,V))$.
Second he points out, that there are PDEs (necessaryly nonlinear), that can't be written in such a way. But I doubt, that the above examples are of this kind.
