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 $\mathfrak{a} \hookrightarrow T X$ 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:
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 $J \subset \Omega^\bullet(X)$ 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 $p^* : J \hookrightarrow \Omega^\bullet(X)$:
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 — see 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 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
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 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 morphism $\phi : \Sigma \to X$ of manifolds such that the pullback of the ideal vanishes, $\phi^* J = 0$.
But this says precisely that $\phi$ extends to morphism of Lie-∞-algebroids
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 Lie-∞-algebroids.
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
$\forall \theta\in J \subset \Omega^\bullet(X), \omega \in \Omega^\bullet(X): \theta \wedge \omega \in J$
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 $J_k = J \cap \Omega^\bullet(k)$
$\forall \theta \in J \subset \Omega^\bullet(X) : d \theta \in J$
An 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: $\that|_Y = 0$.
In other words, for an integral manifold the pullback of the ideal $J$ along the inclusion map $\phi$ vanishes: $\phi^* J = 0$.
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
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.
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).
Some special types of exterior differential systems carry their own names.
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 $E$:
$J = \{\theta \in \Omega^1(X) | \theta|_{E} = 0\}$.
Notice that the involutive subbundle may be thought of precisely as a sub-Lie algebroid
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:
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
and
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.
A system
of partial differential equations in terms of variables $\{x^\mu\}_{\mu = 1}^n$ and functions $\{f^i\}_{i = 1}^s$ of the form
is encoded by an exterior differential system on the 0-locus
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
Namely a solution to the system of partial differential equations is precisely a section of the projection
which defined an integral manifold of the exterior differential system.
The standard textbook is
Course note are provided in
Last revised on July 26, 2020 at 23:34:30. See the history of this page for a list of all contributions to it.