Integrability conditions for differential systems

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, certain systems of partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential form

Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

s. The idea is to take advantage of the way a differential form restricts to a submanifold

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 S → M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required...

, and the fact that this restriction is compatible with the exterior derivative

Exterior derivative

In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....

. This is one possible approach to certain over-determined systems, for example. A Pfaffian system is one specified by 1-forms alone, but the theory includes other types of example of differential system.

Given a collection of differential 1-forms

on an n-dimensional manifold M,
an integral manifold is a submanifold whose tangent space at every point

is annihilated by each

.

A maximal integral manifold is a submanifold

such that the kernel of the restriction map on forms

is spanned by the

at every point p of N. If in addition the

are linearly independent, then N is (n − k)-dimensional. Note that

need not be an embedded submanifold.

A Pfaffian system is said to be completely integrable if N admits a foliation

Foliation

In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....

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

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

Frobenius theorem (differential topology)

In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations...

. One version states that if the ideal

algebraically generated by the collection of

inside the ring

is differentially closed, in other words

then the system admits a foliation

Foliation

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 on

were in the ideal generated by

we would have, by the skewness of the wedge product

But a direct calculation gives

which is a nonzero multiple of the standard volume form on

. Therefore, there are no two-dimensional leaves, and the system is not completely integrable.

On the other hand, the curve defined by

is easily verified to be a solution (i.e. an integral curve

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...

) for the above Pfaffian system for any nonzero constant

Examples of applications

In Riemannian geometry

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 which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

, we may consider the problem of finding an orthogonal coframe

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\bigwedge^kT^*M, given by v_k:\mapsto...

(i.e., collection of 1-forms forming a basis of the cotangent space at every point with

) which are closed

. By the Poincaré lemma, the

locally will have the form

for some functions

on the manifold, and thus provide an isometry of an open subset of M with an open subset of

. Such a manifold is called locally flat.

This problem reduces to a question on the coframe bundle

Frame bundle

In mathematics, a frame bundle is a principal fiber bundle F associated to any vector bundle E. The fiber of F over a point x is the set of all ordered bases, or frames, for Ex...

of M. Suppose we had such a closed coframe

.

If we had another coframe

, then the two coframes would be related by an orthogonal transformation

If the connection 1-form is

, then we have

On the other hand,

But

is the Maurer–Cartan form for the orthogonal group

Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...

. Therefore it obeys the structural equation

and this is just the curvature

Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

of M:

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 which are the Cartan-Kähler theorem, which only works for real analytic

Real analysis

Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...

differential systems, and the Cartan–Kuranishi prolongation theorem. See Further reading for details.