Pullback of an exterior differential system An exterior differential system (EDS) is a pair $(M,\mathcal{E})$ consisting of a smooth manifold $M$ and a homogeneous differentially closed ideal $\mathcal{E}$ of the graded algebra $\Omega^*(M)$ of differential forms on $M$.
An integral manifold of an EDS $\mathcal{E}$ is defined as a submanifold
$$
\iota: N\to M
$$
such that $\iota^*(\varphi)= 0$ for all $\varphi \in \mathcal{E}$
Question: With this definitions in mind, I wonder: if $N$ is a submanifold of $M$ which is not an integral manifold of $\mathcal{E}$, it is at least the set $\iota^*(\mathcal{E})$ such that $(N,\iota^*(\mathcal{E}))$ is an exterior differential system? Or in other words, it is $\iota^*(\mathcal{E})$ a differentially closed ideal of $\Omega^*(N)$? If not, are you aware of a definition of the pullback of an EDS?
 A: If we assume that $\iota$ is a closed map then we can answer in the affirmative. Because $\iota^*(\alpha\wedge\beta)=\iota^*(\alpha)\wedge\iota^*(\beta)$ for all $\alpha,\beta\in \Omega^*(M)$, $\iota^*: \Omega^*(M)\to \Omega^*(N)$ is a ring homomorphism, and as such $\iota^*(\mathcal{E})$ is an ideal of $\iota^*(\Omega^*(M))$. If $\iota$ is a closed map, $\iota^*$ is surjective and $\iota^*(\Omega^*(M))=\Omega^*(N)$. If $\iota^*(\beta)\in \iota^*(\mathcal{E})$ then $d\beta\in \mathcal{E}$ and $d(\iota^*(\beta))=\iota^*(d\beta)\in \iota^*(\mathcal{E})$ and $\iota^*(\mathcal{E})$ is differentially closed.
Since the pullback can fail to be surjective for a non-closed embedding, I don't believe much more can be said in the general case.
When $\iota: N\to M$ is not closed, we see that surjectivity may fail in the following example:
Consider the inclusion of the open interval $(0,1)$ into $\mathbb{R}$. The function $\frac{1}{x}$ is smooth on $(0,1)$, i.e. $\frac{1}{x}\in \Omega^0((0,1))$. However, there is no smooth function $f: \mathbb{R}\to \mathbb{R}$ such that $f\big\vert_{(0,1)}=\frac{1}{x}$. For such an $f$ to exist it must be continuous at $0$, $f(0)=\lim_{x\to 0} f(x)=\lim_{x\to 0^+}\frac{1}{x}$ which cannot be. In general, $\iota^*$ will fail to be surjective due to non-extensible functions like $\frac{1}{x}$.
