Exterior differential system EDS

Definition and notation

Definition (See Bryant_2002 page xii or @bryant2013exterior page 16). 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 ${\mathrm{\Omega }}^{\ast }(M)$ of differential forms on $M$.
Sometimes, an independence condition is required: a differential $n$-form $\mathrm{\Omega }$ is singled out from ${\mathrm{\Omega }}^{\ast }(M)$, in addition to $\mathcal{E}$. It is denoted by $(\mathcal{E},\mathrm{\Omega })$.$◼$

Almost always (see @bryant2013exterior page 13) $\mathcal{E}$ will be generated as an algebraic and differential closure of a finite collection of differential forms (not necessarily of the same degree which, on the other hand, could be 0) ${\alpha }_A$, $1\le A\le N$. This EDSs are called of finite type.

Notation: given a set $S\subset {\mathrm{\Omega }}^{\ast }(M)$, $S_{alg}$ will denote the algebraic ideal generated by the elements of $S$. On the other hand, $S_{diff}$ will denote the ideal generated algebraic-differentially, that is, the differential closure of $S_{alg}$. Then, if $S=\{{\omega }_1,\dots ,{\omega }_k\}$

When the ideal $\mathcal{E}$ is algebraic-differentially generated by a finite set of 1-forms, we say this submodule is a Pfaffian system.

Integral manifolds

Definition. An integral manifold of an EDS $\mathcal{E}$ is a submanifold given by an immersion

such that ${\iota }^{\ast }(\phi )=0$ for all $\phi \in \mathcal{E}$.
If the EDS had an independence condition $\mathrm{\Omega }$, then ${\iota }^{\ast }(\mathrm{\Omega })$ is required to be nonvanishing.

On the other hand, an integral element of a EDS with independence condition $(\mathcal{E},\mathrm{\Omega })$ at a point $x\in M$ is $E\in G(n,T_{x}M)$ such that $\alpha {|}_E=0$ for every $\alpha \in \mathcal{E}$ and $\mathrm{\Omega }{|}_E\ne 0$. They are candidates to be tangent spaces to integral manifolds. The set of $n$-dimensional integral elements at a point $x$ is denoted by ${\mathcal{V}}_n(\mathcal{E}{)}_x$, and it is a subset of the Grassmannian manifold $G(n,T_{x}M)$. This subset is defined by polynomials, since its elements represent the result of evaluating differential forms (see @ivey2016cartan).
The notation ${\mathcal{V}}_n(\mathcal{E})$ is used for the subset of the Grassmannian bundle $G(n,TM)$ consisting of all the integral elements (pairs $(p,E)$).
$◼$

The Cartan-Kähler theory (see Cartan-Kähler theory) studies the existence of integral manifolds by analysing the polar spaces $H(E)$ of a flag of integral elements, culminating in Cartan's test for involutivity.

An important notion is that of Cauchy characteristic vector fields.
It is also important the notion of symmetry of an EDS and symmetry of a Pfaffian system.

Given an immersed submanifold we can define, locally, the pullback or restriction of the EDS. To define it in a global sense we need the immersion to be a closed map. See this MSE question.

Related: EDS vs Pfaffian systems