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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, certain systems of
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
s are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s. The idea is to take advantage of the way a differential form ''restricts'' to a
submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
, and the fact that this restriction is compatible with the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
. This is one possible approach to certain over-determined systems, for example, including
Lax pairs In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the ''Lax equation''. Lax pairs were introduced by Peter Lax to discuss sol ...
of
integrable systems In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...
. A Pfaffian system is specified by
1-form In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ...
s alone, but the theory includes other types of example of differential system. To elaborate, a Pfaffian system is a set of 1-forms on a smooth manifold (which one sets equal to 0 to find ''solutions'' to the system). Given a collection of differential 1-forms \textstyle\alpha_i, i=1,2,\dots, k on an \textstyle n-dimensional manifold M, an integral manifold is an immersed (not necessarily embedded) submanifold whose tangent space at every point \textstyle p\in N is annihilated by (the pullback of) each \textstyle \alpha_i. A maximal integral manifold is an immersed (not necessarily embedded) submanifold :i:N\subset M such that the kernel of the restriction map on forms :i^*:\Omega_p^1(M)\rightarrow \Omega_p^1(N) is spanned by the \textstyle \alpha_i at every point p of N. If in addition the \textstyle \alpha_i are linearly independent, then N is (n-k)-dimensional. A Pfaffian system is said to be completely integrable if M admits a
foliation In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...
by maximal integral manifolds. (Note that the foliation need not be regular; i.e. the leaves of the foliation might not be embedded submanifolds.) An integrability condition is a condition on the \alpha_i to guarantee that there will be integral submanifolds of sufficiently high dimension.


Necessary and sufficient conditions

The necessary and sufficient conditions for complete integrability of a Pfaffian system are given by the Frobenius theorem. One version states that if the ideal \mathcal I algebraically generated by the collection of α''i'' inside the ring Ω(''M'') is differentially closed, in other words :d\subset , then the system admits a
foliation In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...
by maximal integral manifolds. (The converse is obvious from the definitions.)


Example of a non-integrable system

Not every Pfaffian system is completely integrable in the Frobenius sense. For example, consider the following one-form : :\theta=z\,dx +x\,dy+y\,dz. If ''dθ'' were in the ideal generated by ''θ'' we would have, by the skewness of the wedge product :\theta\wedge d\theta=0. But a direct calculation gives :\theta\wedge d\theta=(x+y+z)\,dx\wedge dy\wedge dz which is a nonzero multiple of the standard volume form on R3. Therefore, there are no two-dimensional leaves, and the system is not completely integrable. On the other hand, for the curve defined by : x =t, \quad y= c, \qquad z = e^, \quad t > 0 then θ defined as above is 0, and hence the curve is easily verified to be a solution (i.e. an
integral curve In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations. Name Integral curves are known by various other names, depending on the nature and interpret ...
) for the above Pfaffian system for any nonzero constant ''c''.


Examples of applications

In
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
, we may consider the problem of finding an orthogonal
coframe In mathematics, a coframe or coframe field on a smooth manifold M is a system of one-forms or covectors which form a basis of the cotangent bundle at every point. In the exterior algebra of M, one has a natural map from v_k:\bigoplus^kT^*M\to\big ...
''θ''''i'', i.e., a collection of 1-forms forming a basis of the cotangent space at every point with \langle\theta^i,\theta^j\rangle=\delta^ which are closed (dθ''i'' = 0, ''i'' = 1, 2, ..., ''n''). By the Poincaré lemma, the θ''i'' locally will have the form d''xi'' for some functions ''xi'' on the manifold, and thus provide an isometry of an open subset of ''M'' with an open subset of R''n''. Such a manifold is called locally flat. This problem reduces to a question on the coframe bundle of ''M''. Suppose we had such a closed coframe :\Theta=(\theta^1,\dots,\theta^n). If we had another coframe \Phi=(\phi^1,\dots,\phi^n), then the two coframes would be related by an orthogonal transformation :\Phi=M\Theta If the connection 1-form is ''ω'', then we have :d\Phi=\omega\wedge\Phi On the other hand, : \begin d\Phi & = (dM)\wedge\Theta+M\wedge d\Theta \\ & =(dM)\wedge\Theta \\ & =(dM)M^\wedge\Phi. \end But \omega=(dM)M^ is the
Maurer–Cartan form In mathematics, the Maurer–Cartan form for a Lie group is a distinguished differential one-form on that carries the basic infinitesimal information about the structure of . It was much used by Élie Cartan as a basic ingredient of his met ...
for the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
. Therefore, it obeys the structural equation d\omega+\omega\wedge\omega=0, and this is just the
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
of M: \Omega=d\omega+\omega\wedge\omega=0. After an application of the Frobenius theorem, one concludes that a manifold M is locally flat if and only if its curvature vanishes.


Generalizations

Many generalizations exist to integrability conditions on differential systems which are not necessarily generated by one-forms. The most famous of these are the
Cartan–Kähler theorem In mathematics, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals I. It is named for Élie Cartan and Erich Kähler. Meaning It is no ...
, which only works for
real analytic In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex a ...
differential systems, and the
Cartan–Kuranishi prolongation theorem Given an exterior differential system defined on a manifold ''M'', the Cartan–Kuranishi prolongation theorem says that after a finite number of ''prolongations'' the system is either ''in involution'' (admits at least one 'large' integral mani ...
. See ''Further reading'' for details. The
Newlander-Nirenberg theorem In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not com ...
gives integrability conditions for an almost-complex structure.


Further reading

*Bryant, Chern, Gardner, Goldschmidt, Griffiths, ''Exterior Differential Systems'', Mathematical Sciences Research Institute Publications, Springer-Verlag, *Olver, P., ''Equivalence, Invariants, and Symmetry'', Cambridge, *Ivey, T., Landsberg, J.M., ''Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems'', American Mathematical Society, *Dunajski, M., ''Solitons, Instantons and Twistors'', Oxford University Press, {{ISBN, 978-0-19-857063-9 Partial differential equations Differential topology Differential systems