Frobenius theorem (differential topology)

In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, the theorem gives necessary and sufficient conditions for the existence of a foliation by maximal integral manifolds each of whose tangent bundles are spanned by a given family of vector fields (satisfying an integrability condition) in much the same way as an integral curve may be assigned to a single vector field. The theorem is foundational in differential topology and calculus on manifolds.

Introduction

In its most elementary form, the theorem addresses the problem of finding a maximal set of independent solutions of a regular system of first-order linear homogeneous partial differential equations. Let

\left\{f_{k}^{i}:\mathbf {R} ^{n}\to \mathbf {R} \ :\ 1\leq i\leq n,1\leq k\leq r\right\}

be a collection of $C 1$ functions, with $r < n$ , and such that the matrix $(f i k)$ has rank r. Consider the following system of partial differential equations for a $C 2$ function $u : R n \to R$ :

(1)\quad {\begin{cases}L_{1}u\ {\stackrel {\mathrm {def} }{=}}\ \sum _{i}f_{1}^{i}(x){\frac {\partial u}{\partial x^{i}}}=0\\L_{2}u\ {\stackrel {\mathrm {def} }{=}}\ \sum _{i}f_{2}^{i}(x){\frac {\partial u}{\partial x^{i}}}=0\\\qquad \cdots \\L_{r}u\ {\stackrel {\mathrm {def} }{=}}\ \sum _{i}f_{r}^{i}(x){\frac {\partial u}{\partial x^{i}}}=0\end{cases}}

One seeks conditions on the existence of a collection of solutions $u 1, ..., u n - r$ such that the gradients $\nabla u 1, ..., \nabla u n - r$ are linearly independent.

The Frobenius theorem asserts that this problem admits a solution locally^[1] if, and only if, the operators $L k$ satisfy a certain integrability condition known as involutivity. Specifically, they must satisfy relations of the form

L_{i}L_{j}u(x)-L_{j}L_{i}u(x)=\sum _{k}c_{ij}^{k}(x)L_{k}u(x)

for $1 \leq i, j \leq r$ , and all $C 2$ functions u, and for some coefficients c^k_ij(x) that are allowed to depend on x. In other words, the commutators $[L i, L j]$ must lie in the linear span of the $L k$ at every point. The involutivity condition is a generalization of the commutativity of partial derivatives. In fact, the strategy of proof of the Frobenius theorem is to form linear combinations among the operators $L i$ so that the resulting operators do commute, and then to show that there is a coordinate system $y i$ for which these are precisely the partial derivatives with respect to $y 1, ..., y r$ .

From analysis to geometry

Solutions to underdetermined systems of equations are seldom unique. For example, the system of differential equations

{\begin{cases}{\frac {\partial f}{\partial x}}+{\frac {\partial f}{\partial y}}=0\\{\frac {\partial f}{\partial y}}+{\frac {\partial f}{\partial z}}=0\end{cases}}

clearly permits multiple solutions. Nevertheless, these solutions still have enough structure that they may be completely described. The first observation is that, even if f₁ and f₂ are two different solutions, the level surfaces of f₁ and f₂ must overlap. In fact, the level surfaces for this system are all planes in $R 3$ of the form $x - y + z = C$ , for $C$ a constant. The second observation is that, once the level surfaces are known, all solutions can then be given in terms of an arbitrary function. Since the value of a solution f on a level surface is constant by definition, define a function C(t) by:

f(x,y,z)=C(t){\text{ whenever }}x-y+z=t.

Conversely, if a function $C (t)$ is given, then each function f given by this expression is a solution of the original equation. Thus, because of the existence of a family of level surfaces, solutions of the original equation are in a one-to-one correspondence with arbitrary functions of one variable.

Frobenius' theorem allows one to establish a similar such correspondence for the more general case of solutions of (1). Suppose that $u 1, ..., u n-r$ are solutions of the problem (1) satisfying the independence condition on the gradients. Consider the level sets^[2] of $(u 1, ..., u n-r)$ as functions with values in $R n-r$ . If $v 1, ..., v n-r$ is another such collection of solutions, one can show (using some linear algebra and the mean value theorem) that this has the same family of level sets but with a possibly different choice of constants for each set. Thus, even though the independent solutions of (1) are not unique, the equation (1) nonetheless determines a unique family of level sets. Just as in the case of the example, general solutions u of (1) are in a one-to-one correspondence with (continuously differentiable) functions on the family of level sets.^[3]

The level sets corresponding to the maximal independent solution sets of (1) are called the integral manifolds because functions on the collection of all integral manifolds correspond in some sense to constants of integration. Once one of these constants of integration is known, then the corresponding solution is also known.

Frobenius' theorem in modern language

The Frobenius theorem can be restated more economically in modern language. Frobenius' original version of the theorem was stated in terms of Pfaffian systems, which today can be translated into the language of differential forms. An alternative formulation, which is somewhat more intuitive, uses vector fields.

Formulation using vector fields

In the vector field formulation, the theorem states that a subbundle of the tangent bundle of a manifold is integrable (or involutive) if and only if it arises from a regular foliation. In this context, the Frobenius theorem relates integrability to foliation; to state the theorem, both concepts must be clearly defined.

One begins by noting that an arbitrary smooth vector field $X$ on a manifold $M$ defines a family of curves, its integral curves $u:I\to M$ (for intervals $I$ ). These are the solutions of ${\dot {u}}(t)=X_{u(t)}$ , which is a system of first-order ordinary differential equations, whose solvability is guaranteed by the Picard–Lindelöf theorem. If the vector field $X$ is nowhere zero then it defines a one-dimensional subbundle of the tangent bundle of $M$ , and the integral curves form a regular foliation of $M$ . Thus, one-dimensional subbundles are always integrable.

If the subbundle has dimension greater than one, a condition needs to be imposed. One says that a subbundle $E \subset TM$ of the tangent bundle TM is integrable (or involutive), if, for any two vector fields $X$ and Y taking values in $E$ , the Lie bracket $[X, Y]$ takes values in $E$ as well. This notion of integrability need only be defined locally; that is, the existence of the vector fields $X$ and Y and their integrability need only be defined on subsets of $M$ .

Also, any foliation of M defines a subbundle $E \subset TM$ : If $p\in M$ and $N \subset M$ is the leaf of the foliation passing through p then we take $E_{p}=T_{p}N$ . Again, this definition is purely local: if the foliation is only defined locally then so is E.

Given the above definitions, Frobenius' theorem states that a subbundle $E$ is integrable if and only if it arises from a regular foliation of $M$ .

Differential forms formulation

Let U be an open set in a manifold $M$ , $Ω 1 (U)$ be the space of smooth, differentiable 1-forms on U, and F be a submodule of $Ω 1 (U)$ of rank r, the rank being constant in value over U. The Frobenius theorem states that F is integrable if and only if for every $p$ in $U$ the stalk F_p is generated by r exact differential forms.

Geometrically, the theorem states that an integrable module of $1$ -forms of rank r is the same thing as a codimension-r foliation. The correspondence to the definition in terms of vector fields given in the introduction follows from the close relationship between differential forms and Lie derivatives. Frobenius' theorem is one of the basic tools for the study of vector fields and foliations.

There are thus two forms of the theorem: one which operates with distributions, that is smooth subbundles D of the tangent bundle TM; and the other which operates with subbundles of the graded ring $Ω(M)$ of all forms on M. These two forms are related by duality. If D is a smooth tangent distribution on $M$ , then the annihilator of D, I(D) consists of all forms $\alpha \in \Omega ^{k}(M)$ (for any $k\in \{1,\dots ,\operatorname {dim} M\}$ ) such that

\alpha (v_{1},\dots ,v_{k})=0

for all $v_{1},\dots ,v_{k}\in D$ . The set I(D) forms a subring and, in fact, an ideal in $Ω(M)$ . Furthermore, using the definition of the exterior derivative, it can be shown that I(D) is closed under exterior differentiation (it is a differential ideal) if and only if D is involutive. Consequently, the Frobenius theorem takes on the equivalent form that $I (D)$ is closed under exterior differentiation if and only if D is integrable.

Generalizations

The theorem may be generalized in a variety of ways.

Infinite dimensions

One infinite-dimensional generalization is as follows.^[4] Let $X$ and $Y$ be Banach spaces, and $A \subset X, B \subset Y$ a pair of open sets. Let

F:A\times B\to L(X,Y)

be a continuously differentiable function of the Cartesian product (which inherits a differentiable structure from its inclusion into X × Y) into the space $L (X, Y)$ of continuous linear transformations of $X$ into Y. A differentiable mapping u : A → B is a solution of the differential equation

(1)\quad y'=F(x,y)

if

\forall x\in A:\quad u'(x)=F(x,u(x)).

The equation (1) is completely integrable if for each $(x_{0},y_{0})\in A\times B$ , there is a neighborhood U of x₀ such that (1) has a unique solution $u (x)$ defined on U such that u(x₀)=y₀.

The conditions of the Frobenius theorem depend on whether the underlying field is $R$ or $C$ . If it is R, then assume F is continuously differentiable. If it is $C$ , then assume F is twice continuously differentiable. Then (1) is completely integrable at each point of $A \times B$ if and only if

D_{1}F(x,y)\cdot (s_{1},s_{2})+D_{2}F(x,y)\cdot (F(x,y)\cdot s_{1},s_{2})=D_{1}F(x,y)\cdot (s_{2},s_{1})+D_{2}F(x,y)\cdot (F(x,y)\cdot s_{2},s_{1})

for all $s 1, s 2 \in X$ . Here $D 1$ (resp. $D 2$ ) denotes the partial derivative with respect to the first (resp. second) variable; the dot product denotes the action of the linear operator $F (x, y) \in L (X, Y)$ , as well as the actions of the operators $D 1 F (x, y) \in L (X, L (X, Y))$ and $D 2 F (x, y) \in L (Y, L (X, Y))$ .

Banach manifolds

The infinite-dimensional version of the Frobenius theorem also holds on Banach manifolds.^[5] The statement is essentially the same as the finite-dimensional version.

Let $M$ be a Banach manifold of class at least C². Let $E$ be a subbundle of the tangent bundle of $M$ . The bundle $E$ is involutive if, for each point $p \in M$ and pair of sections $X$ and Y of $E$ defined in a neighborhood of p, the Lie bracket of $X$ and Y evaluated at p, lies in $E p$ :

[X,Y]_{p}\in E_{p}

On the other hand, $E$ is integrable if, for each $p \in M$ , there is an immersed submanifold $φ : N \to M$ whose image contains p, such that the differential of $φ$ is an isomorphism of TN with $φ -1 E$ .

The Frobenius theorem states that a subbundle $E$ is integrable if and only if it is involutive.

Holomorphic forms

The statement of the theorem remains true for holomorphic 1-forms on complex manifolds — manifolds over $C$ with biholomorphic transition functions.^[6]

Specifically, if $\omega ^{1},\dots ,\omega ^{r}$ are r linearly independent holomorphic 1-forms on an open set in $C n$ such that

d\omega ^{j}=\sum _{i=1}^{r}\psi _{i}^{j}\wedge \omega ^{i}

for some system of holomorphic 1-forms $ψ j i, 1 \leq i, j \leq r$ , then there exist holomorphic functions f_i^j and $g i$ such that, on a possibly smaller domain,

\omega ^{j}=\sum _{i=1}^{r}f_{i}^{j}dg^{i}.

This result holds locally in the same sense as the other versions of the Frobenius theorem. In particular, the fact that it has been stated for domains in $C n$ is not restrictive.

Higher degree forms

The statement does not generalize to higher degree forms, although there are a number of partial results such as Darboux's theorem and the Cartan-Kähler theorem.

History

Despite being named for Ferdinand Georg Frobenius, the theorem was first proven by Alfred Clebsch and Feodor Deahna. Deahna was the first to establish the sufficient conditions for the theorem, and Clebsch developed the necessary conditions. Frobenius is responsible for applying the theorem to Pfaffian systems, thus paving the way for its usage in differential topology.

Applications

In classical mechanics, the integrability of a system's constraint equations determines whether the system is Holonomic or Nonholonomic.

Notes

^ Here locally means inside small enough open subsets of $R n$ . Henceforth, when we speak of a solution, we mean a local solution.
^ A level set is a subset of $R n$ corresponding to the locus of:
$(u 1, ..., u n - r) = (c 1, ..., c n - r)$ ,
for some constants $c i$ .
^ The notion of a continuously differentiable function on a family of level sets can be made rigorous by means of the implicit function theorem.
^ Dieudonné, J (1969). Foundations of modern analysis. Academic Press. Chapter 10.9. {{cite book}}: Unknown parameter |nopp= ignored (|no-pp= suggested) (help)
^ Lang, S. (1995). Differential and Riemannian manifolds. Springer-Verlag. Chapter VI: The theorem of Frobenius. ISBN 978-0-387-94338-1. {{cite book}}: Unknown parameter |nopp= ignored (|no-pp= suggested) (help)
^ Kobayashi, Shoshichi and Nomizu, Katsumi (1969). Foundations of Differential Geometry, Vol. 2. Wiley Interscience. Appendix 8. {{cite book}}: Unknown parameter |nopp= ignored (|no-pp= suggested) (help)CS1 maint: multiple names: authors list (link)

References

H. B. Lawson, The Qualitative Theory of Foliations, (1977) American Mathematical Society CBMS Series volume 27, AMS, Providence RI.
Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See theorem 2.2.26.
Clebsch, A. "Ueber die simultane Integration linearer partieller Differentialgleichungen", J. Reine. Angew. Math. (Crelle) 65 (1866) 257-268.
Deahna, F. "Über die Bedingungen der Integrabilitat ....", J. Reine Angew. Math. 20 (1840) 340-350.
Frobenius, G. "Über das Pfaffsche probleme", J. für Reine und Agnew. Math., 82 (1877) 230-315.

[1] Here locally means inside small enough open subsets of $R n$ . Henceforth, when we speak of a solution, we mean a local solution.

[2] A level set is a subset of $R n$ corresponding to the locus of:
$(u 1, ..., u n - r) = (c 1, ..., c n - r)$ ,
for some constants $c i$ .

[3] The notion of a continuously differentiable function on a family of level sets can be made rigorous by means of the implicit function theorem.

[4] Dieudonné, J (1969). Foundations of modern analysis. Academic Press. Chapter 10.9. {{cite book}}: Unknown parameter |nopp= ignored (|no-pp= suggested) (help)

[5] Lang, S. (1995). Differential and Riemannian manifolds. Springer-Verlag. Chapter VI: The theorem of Frobenius. ISBN 978-0-387-94338-1. {{cite book}}: Unknown parameter |nopp= ignored (|no-pp= suggested) (help)

[6] Kobayashi, Shoshichi and Nomizu, Katsumi (1969). Foundations of Differential Geometry, Vol. 2. Wiley Interscience. Appendix 8. {{cite book}}: Unknown parameter |nopp= ignored (|no-pp= suggested) (help)CS1 maint: multiple names: authors list (link)

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