nLab exterior differential system

Idea

There are a couple of 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 $\mathfrak{a} \hookrightarrow T X$ of the tangent Lie algebroid $T X$ of a manifold that is the kernel of a morphism $p \colon T X \longrightarrow \mathfrak{j}$ of $L_\infty$-algebroids:

  $$\mathfrak{a} \coloneqq ker(p) \hookrightarrow T X \overset{p}{\longrightarrow} \mathfrak{j} \mathrlap{\,,}$$

• In the literature (cf. the references below) the term exterior differential system is instead introduced and understood in the context of dg-algebra as a dg-ideal $J \subset \Omega^\bullet(X)$ 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 $p^* \colon J \hookrightarrow \Omega^\bullet(X)$:

  $$J \xhookrightarrow{\phantom{-}p^*\phantom{-}} \Omega^\bullet(X) \longrightarrow coker(p^*) = \Omega^\bullet(X)/J \,.$$

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

  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 $L_\infty$-algebroid of finite type 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 $L_\infty$-algebroids.

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

  for each system of partial differential equations

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

  there is a smooth manifold $X$ and a dg-ideal $J \in \Omega^\bullet(X)$ such that solutions of the system of equations are given by integral manifolds of the exterior differential system 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 smooth map $\phi \colon \Sigma \to X$ of smooth manifolds such that the pullback of the ideal vanishes, $\phi^* J = 0$.

But this says precisely that $\phi$ extends to morphism of $L_\infty$-algebroids:

$$\phi \colon T \Sigma \to \mathfrak{a}$$

with $CE(\mathfrak{a}) = \Omega^\bullet(X)/J$ 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 $L_\infty$-algebroids.

Definition

Definition (exterior differential system)

An 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 $X$.

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

• $\forall \theta\in J \subset \Omega^\bullet(X), \omega \in \Omega^\bullet(X)\colon \theta \wedge \omega \in J$

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

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

Definition

An integral manifold of an exterior differential system is a submanifold $\phi \colon Y \hookrightarrow X$ such that the pullback of all $\theta \in J$ to $Y$ vanishes: $\that|_Y = 0$.

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

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 $\{\theta_k \in \Omega^\bullet(X)\}$ of differential forms such that $J$ is the dg-ideal generated by these, so that

  $$J \;=\; \left\{ \sum_i f_i \theta_i + \sum_j g_j d \theta_j \big\vert f_i, g_j \in C^\infty(X) \right\} \,.$$

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

  Dually in terms of $L_\infty$-algebroids, this assumption means that $J$ is the Chevalley-Eilenberg algebra of an $L_\infty$-algebroid that happens to be just an $L_\infty$-algebra.

• Definition (strict independence condition)

  A 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

  • $\omega$ 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 $\phi : \Sigma \to X$ for $J$ but now such that $\phi^* \omega$ 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 $J \subset \Omega^\bullet(X)$ 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 $E$: $[v,w] \in \Gamma(E) \subset \Gamma(T X)$:

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

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

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

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

$$\array{ E &&\hookrightarrow&& T X \\ & \searrow && \swarrow \\ && X }$$

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

$$CE(E) = \Omega^\bullet(X)/J \,.$$

Vertical tangent Lie algebroid

A special case of a Lie algebroid corresponding to a Frobenius system is the vertical tangent Lie algebroid $T_{vert} Y$ 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 horizontal differential forms

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

and

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

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

$$\big\{ F^\rho \colon \mathbb{R}^n \times \mathbb{R}^s \times \mathbb{R}^{n \cdot s} \longrightarrow \mathbb{R} \}_\rho$$

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

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

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

$$X \coloneqq \big\{ (x,f,p) \in \mathbb{R}^n \times \mathbb{R}^s \times \mathbb{R}^{n s} \big| \forall \rho \colon F^\rho(x,f,p) = 0 \big\}$$

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

$$\theta^i \coloneqq \mathrm{d} f^i - \textstyle{\sum_{\mu=1}^n} p^i_\mu d x^\mu \,.$$

Namely, a solution to the system of PDEs is precisely a section of the projection

$$X \to \mathbb{R}^n$$

which defined an integral manifold of the exterior differential system.

References

The standard textbook:

• Robert L. Bryant, Shiing-Shen Chern, Robert B. Gardner, Hubert L. Goldschmidt, Phillip A. Griffiths: Exterior Differential Systems, Springer (1990) [doi:10.1007/978-1-4613-9714-4]

Introductions and lecture notes:

• Robert L. Bryant: Nine Lectures on Exterior Differential Systems (1999) [pdf]

• Robert L. Bryant: Notes on exterior differential systems [arXiv:1405.3116, alternative:pdf]

• Benjamin McKay: Introduction to exterior differential systems, Banach Center Publications 117 (2019) 45–55 [arXiv:1706.09697 math.DG, doi:10.4064/bc117-2]