There are several different ways to think about differential systems:
the general abstract way which we shall put forward here:
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 inside the deRham dg-algebra of and all concepts there are developed from this perspective.
The existing literature on exterior differential systems is actually a bit unclear about which additional assumptions on are supposed to be a crucial part of the definition. However, in most applications of interest — see the examples below — it turns out that is in fact a semifree dga (over ).
Here we take this as indication that
the reason that the existing literature does present the desired extra assumptions on the dg-ideal 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 -graded commutative dg-algebra. So precisely with that extra condition on 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.
for each system of partial differential equations
there is a space and a dg-ideal such that solutions of the system of equations are given by integral manifolds of the exterior differential system determined by .
The notion of an integral manifold of an exterior differential system is crucial in the theory: in terms of it is a morphism of manifolds such that the pullback of the ideal vanishes, .
But this says precisely that extends to morphism of Lie-∞-algebroids
with 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.
Notice that being a dg-ideal means explicitly that
the -grading on the dg-algebra is induced from that of in that
An integral manifold of an exterior differential system is a submanifold such that the restriction of all to vanishes: .
In other words, for an integral manifold the pullback of the ideal along the inclusion map vanishes: .
Often further assumptions are imposed on exterior differential systems. Here are some:
Often it is assumed that .
A strict independence condition on an exterior differential system is an -form for some such that
is decomposable into a wedge product of 1-forms mod
is pointwise not an element of .
For am exterior differential system with strict independence condition , an integral manifold is now more restrictively an integral manifold for but now such that is a volume form on (i.e. pointwise non-vanishing).
Some special types of exterior differential systems carry their own names.
A Frobenius system is an exterior differential system that is locally generated as a graded-commutative algebra from a set of 1-forms.
Frobenius systems are in bijection with involutive subbundles of the tangent bundle of , i.e. subbundles such that for also the Lie bracket of vector fields of and lands in : :
given a Frobenius system the sections of are defined locally to be the joint kernel of the maps .
given ab involutive subbundle the corresponding Frobenius system is the collection of 1-forms that vanishes on :
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 is the quotient 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 of a map . This corresponds to the subbundle of vertical vector fields on , with respecct to . The corresponding Frobenius system is that of horizontal differential forms
is the dg-algebra of vertical differential forms with respect to .
of partial differential equations in terms of variables and functions of the form
is encoded by an exterior differential system on the 0-locus
of the (assuming that this is a manifold) with the dg-ideal 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