nLab exterior differential system



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 𝔞TX\mathfrak{a} \hookrightarrow T X of the tangent Lie algebroid TXT X of a manifold that is the kernel of a morphism p:TX𝔧p : T X \to \mathfrak{j} of Lie-∞-algebroids:

    𝔞:=ker(p)TXp𝔧 \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 JΩ (X)J \subset \Omega^\bullet(X) inside the deRham dg-algebra of XX and all concepts there are developed from this perspective.

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

    Jp *Ω (X)coker(p *)=Ω (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 assumptions on JJ are supposed to be a crucial part of the definition. However, in most applications of interest — see the examples below — it turns out that JJ is in fact a semifree dga (over C (X)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 JJ is required to be a semifree dga;

    • the reason that the existing literature does present the desired extra assumptions on the dg-ideal JJ 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 JJ 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 ρ({x μ} μ,{f j} j,{f jx μ})=0} ρ \{ F^\rho(\{x^\mu\}_{\mu}, \{f^j\}_j, \{\frac{\partial f^j}{\partial x^\mu}\} ) = 0 \}_\rho

    there is a space XX and a dg-ideal JΩ (X)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 JJ.

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

But this says precisely that ϕ\phi extends to morphism of Lie-∞-algebroids

ϕ:TΣ𝔞 \phi : T \Sigma \to \mathfrak{a}

with CE(𝔞)=Ω (X)/JCE(\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 Lie-∞-algebroids.


Definition (exterior differential system)

An exterior differential system on a smooth manifold XX is a dg-ideal JΩ (X)J \subset \Omega^\bullet(X) of the deRham dg-algebra Ω (X)\Omega^\bullet(X) of XX.

Notice that JJ being a dg-ideal means explicitly that

  • θJΩ (X),ωΩ (X):θωJ\forall \theta\in J \subset \Omega^\bullet(X), \omega \in \Omega^\bullet(X): \theta \wedge \omega \in J

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

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


An integral manifold of an exterior differential system is a submanifold ϕ:YX\phi : Y \hookrightarrow X such that the restriction of all θJ\theta \in J to YY vanishes: that| Y=0\that|_Y = 0.

In other words, for an integral manifold the pullback of the ideal JJ along the inclusion map ϕ\phi vanishes: ϕ *J=0\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 {θ kΩ (X)}\{\theta_k \in \Omega^\bullet(X)\} of differential forms such that JJ is the dg-ideal generated by these, so that

    J={ if iθ i+ jg jdθ j|f i,g jC (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=J_0 = \mathbb{R}.

    Dually in terms of Lie-∞-algebroids this assumption means that JJ 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 JΩ (X)J \subset \Omega^\bullet(X) is an nn-form ωΩ n(X)\omega \in \Omega^n(X) for some nn such that

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

    • ω\omega is pointwise not an element of JJ.

    For (J,ω)(J, \omega) am exterior differential system with strict independence condition ω\omega, an integral manifold is now more restrictively an integral manifold ϕ:ΣX\phi : \Sigma \to X for JJ but now such that ϕ *ω\phi^* \omega is a volume form on Σ\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 JΩ (X)J \subset \Omega^\bullet(X) that is locally generated as a graded-commutative algebra from a set {θ jΩ 1(U)} j\{\theta_j \in \Omega^1(U)\}_j of 1-forms.

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

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

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

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

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

E TX X \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 EE is the quotient Ω (X)/J\Omega^\bullet(X)/J of the deRham dg-algebra by the Frobenius system:

CE(E)=Ω (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 vertYT_{vert} Y of a map π:YX\pi : Y \to X. This corresponds to the subbundle ker(π *)TYker(\pi_*) \subset T Y of vertical vector fields on YY, with respecct to π\pi. The corresponding Frobenius system is that of horizontal differential forms

J=Ω hor (Y)={ωΩ 1(Y)|vker(π *):ω(v)=0} J = \Omega^\bullet_{hor}(Y) = \{\omega \in \Omega^1(Y)| \forall v \in ker(\pi_*): \omega(v) = 0\}


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

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

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

systems of partial differential equations

A system

{F ρ: n× s× ns} ρ \{ 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 μ} μ=1 n\{x^\mu\}_{\mu = 1}^n and functions {f i} i=1 s\{f^i\}_{i = 1}^s of the form

{F ρ((x μ),(f i),(f ix μ))=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) n× s× ns|ρ:F ρ(x,f,p)=0} X := \{(x,f,p) \in \mathbb{R}^n \times \mathbb{R}^s \times \mathbb{R}^{n s} | \forall \rho : F^\rho(x,f,p) = 0 \}

of the {F ρ} ρ\{F^\rho\}_\rho (assuming that this is a manifold) with the dg-ideal J=θ i iJ = \langle \theta_i \rangle_i generated by the 1-forms

θ i:=df i μ=1 np μ idx μ. \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 n X \to \mathbb{R}^n

which defined an integral manifold of the exterior differential system.


The standard textbook is

  • Bryant et al., Exterior differential systems

Course note are provided in

Last revised on July 27, 2020 at 03:34:30. See the history of this page for a list of all contributions to it.