nLab
exterior differential system

Idea

There are several different ways to think about differential systems:

• the general abstract way which we shall put forward here:

  an exterior differential system is a sub-Lie-∞-algebroid $𝔞\hookrightarrow TX$ of the tangent Lie algebroid $TX$ of a manifold that is the kernel of a morphism $p:TX\to 𝔧$ of Lie-∞-algebroids:

  $$𝔞:=\ker (p)\hookrightarrow TX\stackrel{p}{\to }𝔧$$\mathfrak{a} := ker(p) \hookrightarrow T X \stackrel{p}{\to} \mathfrak{j}

• In the literature – the the references below – the term exterior differential system is instead introduced and understood in the context of dg-algebra as a dg-ideal $J\subset {\Omega }^{•}(X)$ 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 ${\Omega }^{•}(X)/J$ which is the cokernel of the inclusion $p^{*}:J\hookrightarrow {\Omega }^{•}(X)$:

  $$J\stackrel{p^{*}}{\hookrightarrow }{\Omega }^{•}(X)\to \mathrm{coker}(p^{*})={\Omega }^{•}(X)/J.$$J \stackrel{p^*}{\hookrightarrow} \Omega^\bullet(X) \to coker(p^*) = \Omega^\bullet(X)/J \,.

  The existing literature on exterior differential systems is actually a bit unclear about which additional assumptionsurther assumptions) on $J$ are supposed to be a crucial part of the definition. However, in most applications of interest — see the examples below — it turns out that $J$ is in fact a semifree dga (over $C^{\mathrm{\infty }}(X)$).

  Here we take this as indication that

  • 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;

  • 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.

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

• Historically, one can trace back the basic idea of exterior differential systems to Eli Cartan’s work on partial differential equations in terms of differential forms:

  for each system of partial differential equations

  $$\{F^{\rho }(\{x^{\mu }{\}}_{\mu },\{f^j{\}}_j,\{\frac{\partial f^j}{\partial x^{\mu }}\})=0{\}}_{\rho }$$\{ F^\rho(\{x^\mu\}_{\mu}, \{f^j\}_j, \{\frac{\partial f^j}{\partial x^\mu}\} ) = 0 \}_\rho

  there is a space $X$ and a dg-ideal $J\in {\Omega }^{•}(X)$ such that solutions of the system of equations are given by integral manifolds of the exterior differential syetem determined by $J$.

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

But this says precisely that $\varphi$ extends to morphism of Lie-∞-algebroids

$$\varphi :T\Sigma \to 𝔞$$\phi : T \Sigma \to \mathfrak{a}

with $\mathrm{CE}(𝔞)={\Omega }^{•}(X)/J$ as above. Therefore the relevance of the notion of integral manifolds in the theory we take as another indication that exterior differential systems are usefully thought of as being about Lie-∞-algebroids.

Definition

Definition (exterior differential system)

An exterior differential system on a smooth manifold $X$ is a dg-ideal $J\subset {\Omega }^{•}(X)$ of the deRham dg-algebra ${\Omega }^{•}(X)$ of $X$.

Notice that $J$ being a dg-ideal means explicitly that

• $\forall \theta \in J\subset {\Omega }^{•}(X),\omega \in {\Omega }^{•}(X):\theta \wedge \omega \in J$

• the $\mathbb{N}$-grading $J={\oplus }_{k\in \mathbb{N}}{J}_k$ on the dg-algebra $J$ is induced from that of ${\Omega }^{•}(X)$ in that $J_k=J\cap {\Omega }^{•}(k)$

• $\forall \theta \in J\subset {\Omega }^{•}(X):d\theta \in J$

Definition

An integral manifold of an exterior differential system is a submanifold $\varphi :Y\hookrightarrow X$ such that the restriction of all $\theta \in J$ to $Y$ vanishes: $that{\mid }_Y=0$.

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

common further assumptions

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

• An exterior differential system is called finitely generated if there is a finite set $\{{\theta }_k\in {\Omega }^{•}(X)\}$ of differential forms such that $J$ is the dg-ideal generated by these, so that

  $$J=\{\sum _i{f}_i{\theta }_i+\sum _j{g}_{j}d{\theta }_j\mid f_i,g_j\in C^{\mathrm{\infty }}(X)\}.$$J = \{ \sum_i f_i \theta_i + \sum_j g_j d \theta_j| f_i, g_j \in C^\infty(X)\} \,.

• Often it is assumed that $J_0=ℝ$.

  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.

• Definition (strict independence condition)

  A strict independence condition on an exterior differential system $J\subset {\Omega }^{•}(X)$ is an $n$-form $\omega \in {\Omega }^n(X)$ for some $n$ such that

  • $\omega$ is decomposable into a wedge product of $n$ 1-forms mod $J^n$

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

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

special cases

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

Frobenius system

A Frobenius system is an exterior differential system $J\subset {\Omega }^{•}(X)$ 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 $E\hookrightarrow TX$ such that for $v,w\in \Gamma (E)\subset \Gamma (TX)$ also the Lie bracket of vector fields of $v$ and $w$ lands in $E$: $[v,w]\in \Gamma (E)\subset \Gamma (TX)$:

• given a Frobenius system the sections of $\Gamma (E)$ are defined locally to be the joint kernel of the maps $\{{\theta }_i:\Gamma (TU)\to ℝ\}$.

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

  $J=\{\theta \in {\Omega }^1(X)\mid \theta {\mid }_E=0\}$.

Notice that the involutive subbundle may be thought of precisely as a sub-Lie algebroid

$$\begin{array}{ccccc}E& & \hookrightarrow & & TX\\ & \searrow & & \swarrow \\ & & X\end{array}$$\array{ E &&\hookrightarrow&& T X \\ & \searrow && \swarrow \\ && X }

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 ${\Omega }^{•}(X)/J$ of the deRham dg-algebra by the Frobenius system:

$$\mathrm{CE}(E)={\Omega }^{•}(X)/J.$$CE(E) = \Omega^\bullet(X)/J \,.

vertical tangent Lie algebroid

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

$$J={\Omega }_{\mathrm{hor}}^{•}(Y)=\{\omega \in {\Omega }^1(Y)\mid \forall v\in \ker ({\pi }_{*}):\omega (v)=0\}$$J = \Omega^\bullet_{hor}(Y) = \{\omega \in \Omega^1(Y)| \forall v \in ker(\pi_*): \omega(v) = 0\}

and

$$\mathrm{CE}(T_{\mathrm{vert}}Y)={\Omega }^{•}(Y)/{\Omega }_{\mathrm{hor}}^{•}(Y)$$CE(T_{vert} Y) = \Omega^\bullet(Y)/\Omega^\bullet_{hor}(Y)

is the dg-algebra of vertical differential forms with respect to $Y$.

This plays a central role in the theory of Ehresmann connections and Cartan-Ehresmann ∞-connection.

systems of partial differential equations

A system

$$\{F^{\rho }:{ℝ}^n\times {ℝ}^s\times {ℝ}^{n\cdot s}\to ℝ{\}}_{\rho }$$\{ F^\rho : \mathbb{R}^n \times \mathbb{R}^s \times \mathbb{R}^{n \cdot s} \to \mathbb{R} \}_\rho

of partial differential equations in terms of variables $\{x^{\mu }{\}}_{\mu =1}^n$ and functions $\{f^i{\}}_{i=1}^s$ of the form

$$\{F^{\rho }((x^{\mu }),(f^i),\left(\frac{\partial f^i}{\partial x^{\mu }}\right))=0\}$$\{ F^\rho((x^\mu), (f^i), \left(\frac{\partial f^i}{\partial x^\mu}\right)) = 0 \}

is encoded by an exterior differential system on the 0-locus

$$X:=\{(x,f,p)\in {ℝ}^n\times {ℝ}^s\times mthbb{R}^{ns}\mid \forall \rho :F^{\rho }(x,f,p)=0\}$$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 \}

of the $\{F^{\rho }{\}}_{\rho }$ (assuming that this is a manifold) with the dg-ideal $J=⟨{\theta }_i{⟩}_i$ generated by the 1-forms

$${\theta }^i:=d{f}^i-\sum _{\mu =1}^n{p}_{\mu }^{i}d{x}^{\mu }.$$\theta^i := d f^i - \sum_{\mu=1}^n p^i_\mu d x^\mu \,.

Namely a solution to the system of partial differential equations is precisely a section of the projection

$$X\to {ℝ}^n$$X \to \mathbb{R}^n

which defined an integral manifold of the exterior differential system.

References

The standard textbook is

• Bryant et al., Exterior differential systems

Course note are provided in

• Bryant, Notes on Exterior Differential Systems