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\begin{document}

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\section*{exterior differential system}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{contents}{Contents}\dotfill \pageref*{contents} \linebreak
\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{commonfurtherassumptions}{Common further assumptions}\dotfill \pageref*{commonfurtherassumptions} \linebreak
\noindent\hyperlink{special_cases}{special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{frobenius_system}{Frobenius system}\dotfill \pageref*{frobenius_system} \linebreak
\noindent\hyperlink{vertical_tangent_lie_algebroid}{vertical tangent Lie algebroid}\dotfill \pageref*{vertical_tangent_lie_algebroid} \linebreak
\noindent\hyperlink{systems_of_partial_differential_equations}{systems of partial differential equations}\dotfill \pageref*{systems_of_partial_differential_equations} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\section*{{Idea}}\label{idea}

There are several different ways to think about differential systems:

\begin{itemize}%
\item the [[category theory|general abstract way]] which we shall put forward here:

an exterior differential system is a sub-[[Lie-∞-algebroid]] $\mathfrak{a} \hookrightarrow T X$ of the [[tangent Lie algebroid]] $T X$ of a [[manifold]] that is the [[kernel]] of a morphism $p : T X \to \mathfrak{j}$ of [[Lie-∞-algebroids]]:

\begin{displaymath}
\mathfrak{a} := ker(p) \hookrightarrow T X \stackrel{p}{\to}
    \mathfrak{j}
\end{displaymath}

\item In the literature -- the the \hyperlink{references}{references} below -- the term \emph{exterior differential system} is instead introduced and understood in the context of [[differential graded algebra|dg-algebra]] as a [[dg-ideal]] $J \subset \Omega^\bullet(X)$ inside the [[deRham dg-algebra]] of $X$ and all concepts there are developed from this perspective.

From this the above perspective is obtained by noticing that from a dg-ideal $J$ we are naturally led to form the quotient [[dg-algebra]] $\Omega^\bullet(X)/J$ which is the [[cokernel]] of the inclusion $p^* : J \hookrightarrow \Omega^\bullet(X)$:

\begin{displaymath}
J \stackrel{p^*}{\hookrightarrow} \Omega^\bullet(X) \to coker(p^*) = 
  \Omega^\bullet(X)/J
  \,.
\end{displaymath}
The existing literature on exterior differential systems is actually a bit unclear about which \hyperlink{commonfurtherassumptions}{additional assumptions} on $J$ are supposed to be a crucial part of the definition. However, in most applications of interest --- see the \hyperlink{examples}{examples} below --- it turns out that $J$ is in fact a [[semifree dga]] (over $C^\infty(X)$).

Here we take this as indication that

\begin{itemize}%
\item it makes good sense to understand exterior differential systems in the restricted sense where the [[dg-ideal]] $J$ is required to be a [[semifree dga]];


\item the reason that the existing literature does present the desired extra assumptions on the [[dg-ideal]] $J$ in an incoherent fashion is due to a lack of global structural insight into the role of the definition of exterior differential systems.



\end{itemize}
Because, recall that a [[Lie-∞-algebroid]] is -- effectively by definition -- the formal dual of a [[semifree dga|semifree]] $\mathbb{N}$-graded commutative [[dg-algebra]]. So precisely with that extra condition on $J$ all [[dg-algebra]]s in the above may be understood as [[Chevalley-Eilenberg algebra]]s of [[Lie-∞-algebroids]] and then the above [[cokernel]] sequence of [[dg-algebra]]s is precisely the formal dual of the [[kernel]] sequence of [[Lie-∞-algebroid]]s.


\item Historically, one can trace back the basic idea of exterior differential systems to [[Eli Cartan]]`s work on [[partial differential equation]]s in terms of [[differential forms]]:

for each system of partial differential equations

\begin{displaymath}
\{
    F^\rho(\{x^\mu\}_{\mu}, \{f^j\}_j, \{\frac{\partial f^j}{\partial x^\mu}\} ) = 0
  \}_\rho
\end{displaymath}
there is a space $X$ and a [[dg-ideal]] $J \in \Omega^\bullet(X)$ such that solutions of the system of equations are given by \textbf{integral manifolds} of the exterior differential system determined by $J$.



\end{itemize}
The notion of an integral manifold of an exterior differential system is crucial in the theory: in terms of $J$ it is a morphism $\phi : \Sigma \to X$ of [[manifold]]s such that the pullback of the ideal vanishes, $\phi^* J = 0$.

But this says precisely that $\phi$ extends to morphism of [[Lie-∞-algebroids]]

\begin{displaymath}
\phi : T \Sigma \to \mathfrak{a}
\end{displaymath}
with $CE(\mathfrak{a}) = \Omega^\bullet(X)/J$ as above. Therefore the relevance of the notion of \emph{integral manifolds} in the theory we take as another indication that exterior differential systems are usefully thought of as being about [[Lie-∞-algebroids]].

\hypertarget{definition}{}\section*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
An \textbf{exterior differential system} on a smooth [[manifold]] $X$ is a [[dg-ideal]] $J \subset \Omega^\bullet(X)$ of the [[deRham dg-algebra]] $\Omega^\bullet(X)$ of $X$.

\end{defn}
Notice that $J$ being a [[dg-ideal]] means explicitly that

\begin{itemize}%
\item $\forall \theta\in J \subset \Omega^\bullet(X), \omega \in \Omega^\bullet(X): 
\theta \wedge \omega \in J$


\item the $\mathbb{N}$-grading $J = \oplus_{k \in \mathbb{N}} J_k$ on the [[dg-algebra]] $J$ is induced from that of $\Omega^\bullet(X)$ in that $J_k = J \cap \Omega^\bullet(k)$


\item $\forall \theta \in J \subset \Omega^\bullet(X) : d \theta \in J$



\end{itemize}
\begin{defn}
\label{}\hypertarget{}{}
An \textbf{integral manifold} of an exterior differential system is a submanifold $\phi : Y \hookrightarrow X$ such that the restriction of all $\theta \in J$ to $Y$ vanishes: $\that|_Y = 0$.

\end{defn}
In other words, for an integral manifold the pullback of the ideal $J$ along the inclusion map $\phi$ vanishes: $\phi^* J = 0$.

\hypertarget{commonfurtherassumptions}{}\subsection*{{Common further assumptions}}\label{commonfurtherassumptions}

Often further assumptions are imposed on exterior differential systems. Here are some:

\begin{itemize}%
\item An exterior differential system is called \textbf{finitely generated} if there is a finite set $\{\theta_k \in \Omega^\bullet(X)\}$ of [[differential form]]s such that $J$ is the [[dg-ideal]] generated by these, so that

\begin{displaymath}
J = \{ \sum_i f_i \theta_i  + \sum_j g_j d \theta_j| f_i, g_j \in C^\infty(X)\}
  \,.
\end{displaymath}

\item Often it is assumed that $J_0 = \mathbb{R}$.

Dually in terms of [[Lie-∞-algebroids]] this assumption means that $J$ is the [[Chevalley-Eilenberg algebra]] of a [[Lie-∞-algebroid]] that is just an [[L-∞-algebra]].


\item \begin{defn}
\label{}\hypertarget{}{}
A \textbf{strict independence condition} on an exterior differential system $J \subset \Omega^\bullet(X)$ is an $n$-form $\omega \in \Omega^n(X)$ for some $n$ such that

\begin{itemize}%
\item $\omega$ is decomposable into a wedge product of $n$ 1-forms mod $J^n$


\item $\omega$ is pointwise not an element of $J$.



\end{itemize}
For $(J, \omega)$ am exterior differential system with strict independence condition $\omega$, an \textbf{integral manifold} is now more restrictively an integral manifold $\phi : \Sigma \to X$ for $J$ but now such that $\phi^* \omega$ is a volume form on $\Sigma$ (i.e. pointwise non-vanishing).

\end{defn}


\end{itemize}
\hypertarget{special_cases}{}\section*{{special cases}}\label{special_cases}

Some special types of exterior differential systems carry their own names.

\hypertarget{frobenius_system}{}\subsection*{{Frobenius system}}\label{frobenius_system}

A \textbf{Frobenius system} is an exterior differential system $J \subset \Omega^\bullet(X)$ that is locally generated as a graded-commutative algebra from a set $\{\theta_j \in \Omega^1(U)\}_j$ of 1-forms.

Frobenius systems are in bijection with involutive subbundles of the [[tangent bundle]] of $X$, i.e. subbundles $E \hookrightarrow T X$ such that for $v,w \in \Gamma(E) \subset \Gamma(T X)$ also the Lie bracket of vector fields of $v$ and $w$ lands in $E$: $[v,w] \in \Gamma(E) \subset \Gamma(T X)$:

\begin{itemize}%
\item given a Frobenius system the sections of $\Gamma(E)$ are defined locally to be the joint [[kernel]] of the maps $\{\theta_i : \Gamma(T U) \to \mathbb{R}\}$.


\item given ab involutive subbundle $E$ the corresponding Frobenius system is the collection of 1-forms that vanishes on $E$:

$J = \{\theta \in \Omega^1(X) | \theta|_{E} = 0\}$.



\end{itemize}
Notice that the involutive subbundle may be thought of precisely as a sub-[[Lie algebroid]]

\begin{displaymath}
\itexarray{
     E &&\hookrightarrow&& T X
     \\
     & \searrow && \swarrow
     \\
     && X
  }
\end{displaymath}
of the [[tangent Lie algebroid]] (i.e. as a sub [[Lie-∞-algebroid]] that happens to be an ordinary [[Lie algebroid]]). And indeed, the [[Chevalley-Eilenberg algebra]] of $E$ is the quotient $\Omega^\bullet(X)/J$ of the [[deRham dg-algebra]] by the Frobenius system:

\begin{displaymath}
CE(E) = \Omega^\bullet(X)/J
  \,.
\end{displaymath}
\hypertarget{vertical_tangent_lie_algebroid}{}\subsubsection*{{vertical tangent Lie algebroid}}\label{vertical_tangent_lie_algebroid}

A special case of a [[Lie algebroid]] corresponding to a Frobenius system is the [[vertical tangent Lie algebroid]] $T_{vert} Y$ of a map $\pi : Y \to X$. This corresponds to the subbundle $ker(\pi_*) \subset T Y$ of vertical vector fields on $Y$, with respecct to $\pi$. The corresponding Frobenius system is that of \textbf{horizontal differential forms}

\begin{displaymath}
J = \Omega^\bullet_{hor}(Y) = \{\omega \in \Omega^1(Y)| \forall v \in ker(\pi_*): \omega(v) = 0\}
\end{displaymath}
and

\begin{displaymath}
CE(T_{vert} Y) = \Omega^\bullet(Y)/\Omega^\bullet_{hor}(Y)
\end{displaymath}
is the [[dg-algebra]] of \textbf{vertical differential forms} with respect to $Y$.

This plays a central role in the theory of [[Ehresmann connection]]s and [[schreiber:Cartan-Ehresmann ∞-connection|Cartan-Ehresmann ∞-connection]].

\hypertarget{systems_of_partial_differential_equations}{}\subsection*{{systems of partial differential equations}}\label{systems_of_partial_differential_equations}

A system

\begin{displaymath}
\{
    F^\rho : \mathbb{R}^n \times \mathbb{R}^s \times \mathbb{R}^{n \cdot s}
    \to 
    \mathbb{R}
  \}_\rho
\end{displaymath}
of [[partial differential equation]]s in terms of variables $\{x^\mu\}_{\mu = 1}^n$ and functions $\{f^i\}_{i = 1}^s$ of the form

\begin{displaymath}
\{
    F^\rho((x^\mu), (f^i), \left(\frac{\partial f^i}{\partial x^\mu}\right))
    = 0
  \}
\end{displaymath}
is encoded by an exterior differential system on the 0-locus

\begin{displaymath}
X := \{(x,f,p) \in \mathbb{R}^n \times \mathbb{R}^s \times \mthbb{R}^{n s} |
    \forall \rho : F^\rho(x,f,p) = 0 \}
\end{displaymath}
of the $\{F^\rho\}_\rho$ (assuming that this is a manifold) with the [[dg-ideal]] $J = \langle \theta_i \rangle_i$ generated by the 1-forms

\begin{displaymath}
\theta^i := d f^i - \sum_{\mu=1}^n p^i_\mu d x^\mu
  \,.
\end{displaymath}
Namely a solution to the system of partial differential equations is precisely a [[section]] of the projection

\begin{displaymath}
X \to \mathbb{R}^n
\end{displaymath}
which defined an integral manifold of the exterior differential system.

\hypertarget{references}{}\section*{{References}}\label{references}

The standard textbook is

\begin{itemize}%
\item Bryant et al., \emph{Exterior differential systems}

\end{itemize}
Course note are provided in

\begin{itemize}%
\item Bryant, [[EDS-notes.pdf:file]]

\end{itemize}
[[!redirects exterior differential systems]]



\end{document}