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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 ...
, Frobenius' theorem gives
necessary and sufficient condition In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
s for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s. In modern geometric terms, given a family of vector fields, the theorem gives necessary and sufficient
integrability condition In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...
s for the existence of 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 manifold In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of ...
s whose tangent bundles are spanned by the given vector fields. The theorem generalizes the
existence theorem In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase " there exist(s)", or it might be a universal statement whose last quantifier is existential ...
for ordinary differential equations, which guarantees that a single vector field always gives rise to
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 ...
s; Frobenius gives compatibility conditions under which the integral curves of ''r'' vector fields mesh into coordinate grids on ''r''-dimensional integral manifolds. The theorem is foundational in
differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
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 equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s. Let : \left \ be a collection of functions, with , and such that the matrix has
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
''r''. Consider the following system of partial differential equations for a function : :(1) \quad \begin L_1u\ \stackrel\ \sum_i f_1^i(x)\frac = 0\\ L_2u\ \stackrel\ \sum_i f_2^i(x)\frac = 0\\ \qquad \cdots \\ L_ru\ \stackrel\ \sum_i f_r^i(x)\frac = 0 \end One seeks conditions on the existence of a collection of solutions such that the gradients are
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
. The Frobenius theorem asserts that this problem admits a solution locally if, and only if, the operators satisfy a certain
integrability condition In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...
known as ''involutivity''. Specifically, they must satisfy relations of the form :L_iL_ju(x)-L_jL_iu(x)=\sum_k c_^k(x)L_ku(x) for , and all functions ''u'', and for some coefficients ''c''''k''''ij''(''x'') that are allowed to depend on ''x''. In other words, the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
s must lie in the linear span of the 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 so that the resulting operators do commute, and then to show that there is a
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
for which these are precisely the partial derivatives with respect to .


From analysis to geometry

Even though the system is overdetermined there are typically infinitely many solutions. For example, the system of differential equations :\begin \frac + \frac =0\\ \frac+ \frac=0 \end clearly permits multiple solutions. Nevertheless, these solutions still have enough structure that they may be completely described. The first observation is that, even if ''f''1 and ''f''2 are two different solutions, the level surfaces of ''f''1 and ''f''2 must overlap. In fact, the level surfaces for this system are all planes in of the form , for 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 x - y + z = t. Conversely, if a function 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 are solutions of the problem (1) satisfying the independence condition on the gradients. Consider the
level set In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is calle ...
s of as functions with values in . If is another such collection of solutions, one can show (using some
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
and the
mean value theorem In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...
) 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. 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 system In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...
s, which today can be translated into the language 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 applications, ...
s. 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 In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
is integrable (or involutive) if and only if it arises from a
regular foliation The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrumen ...
. 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
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
s, its integral curves u:I\to M (for intervals I). These are the solutions of \dot u(t) = X_, which is a system of first-order
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
s, whose solvability is guaranteed by the
Picard–Lindelöf theorem In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauc ...
. 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 In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
TM is integrable (or involutive), if, for any two vector fields X and Y taking values in E, the
Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
,Y/math> 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. Several definitions of
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 ...
exist. Here we use the following: Definition. A ''p''-dimensional, class ''Cr'' foliation of an ''n''-dimensional manifold ''M'' is a decomposition of ''M'' into a union of disjoint connected submanifolds α∈''A'', called the ''leaves'' of the foliation, with the following property: Every point in ''M'' has a neighborhood ''U'' and a system of local, class ''Cr'' coordinates ''x''=(''x''1, ⋅⋅⋅, ''xn'') : ''U''→R''n'' such that for each leaf ''L''α, the components of ''U'' ∩ ''L''α are described by the equations ''x''''p''+1=constant, ⋅⋅⋅, ''xn''=constant. A foliation is denoted by \mathcal=α∈''A''. Trivially, any
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 ...
of M defines an integrable subbundle, since if p\in M and N\subset M is the leaf of the foliation passing through p then E_p = T_pN is integrable. Frobenius' theorem states that the converse is also true: Given the above definitions, Frobenius' theorem states that a subbundle E is integrable if and only if the subbundle E arises from a regular foliation of M.


Differential forms formulation

Let ''U'' be an open set in a manifold , be the space of smooth, differentiable
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 ea ...
s on ''U'', and ''F'' be a
submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mod ...
of of
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
''r'', the rank being constant in value over ''U''. The Frobenius theorem states that ''F'' is
integrable 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 ...
if and only if for every in the stalk ''Fp'' is generated by ''r''
exact differential form In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another ...
s. Geometrically, the theorem states that an integrable module of -forms of rank ''r'' is the same thing as a codimension-r
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 ...
. The correspondence to the definition in terms of vector fields given in the introduction follows from the close relationship between
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 applications, ...
s and
Lie derivative In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
s. 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
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...
s, that is smooth subbundles ''D'' of the tangent bundle ''TM''; and the other which operates with subbundles of the graded ring of all forms on ''M''. These two forms are related by duality. If ''D'' is a smooth tangent distribution on , then the annihilator of ''D'', ''I''(''D'') consists of all forms \alpha\in\Omega^k (M) (for any k\in \) 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 . Furthermore, using the definition of 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 res ...
, it can be shown that ''I''(''D'') is closed under exterior differentiation (it is a
differential ideal In the theory of differential forms, a differential ideal ''I'' is an ''algebraic ideal'' in the ring of smooth differential forms on a smooth manifold, in other words a graded ideal in the sense of ring theory, that is further closed under exter ...
) if and only if ''D'' is involutive. Consequently, the Frobenius theorem takes on the equivalent form that 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. Let and be
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s, and a pair of
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
s. Let :F:A\times B \to L(X,Y) be a
continuously differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
of the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
(which inherits a
differentiable structure In mathematics, an ''n''-dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for dif ...
from its inclusion into ''X'' × ''Y'') into the space of
continuous linear transformation In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear ...
s of 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 ''x''0 such that (1) has a unique solution defined on ''U'' such that ''u''(''x''0)=''y''0. The conditions of the Frobenius theorem depend on whether the underlying
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
is or . If it is R, then assume ''F'' is continuously differentiable. If it is , then assume ''F'' is twice continuously differentiable. Then (1) is completely integrable at each point of if and only if :D_1F(x,y)\cdot(s_1,s_2) + D_2F(x,y)\cdot(F(x,y)\cdot s_1,s_2) = D_1F(x,y) \cdot (s_2,s_1) + D_2F(x,y)\cdot(F(x,y)\cdot s_2,s_1) for all . Here (resp. ) denotes the partial derivative with respect to the first (resp. second) variable; the dot product denotes the action of the linear operator , as well as the actions of the operators and .


Banach manifolds

The infinite-dimensional version of the Frobenius theorem also holds on
Banach manifold In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space (a more involved and formal definition is given below). B ...
s. The statement is essentially the same as the finite-dimensional version. Let be a Banach manifold of class at least ''C''2. Let be a subbundle of the tangent bundle of . The bundle is involutive if, for each point and pair of sections and ''Y'' of defined in a neighborhood of ''p'', the Lie bracket of and ''Y'' evaluated at ''p'', lies in : : ,Yp \in E_p On the other hand, is integrable if, for each , there is an immersed submanifold whose image contains ''p'', such that the differential of is an isomorphism of ''TN'' with . The Frobenius theorem states that a subbundle is integrable if and only if it is involutive.


Holomorphic forms

The statement of the theorem remains true for holomorphic 1-forms on
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
s — manifolds over with biholomorphic
transition functions In mathematics, a transition function may refer to: * a transition map between two charts of an atlas of a manifold or other topological space * the function that defines the transitions of a state transition system in computing, which may refer mo ...
. Specifically, if \omega^1,\dots,\omega^r are ''r'' linearly independent holomorphic 1-forms on an open set in such that :d\omega^j = \sum_^r \psi_i^j \wedge \omega^i for some system of holomorphic 1-forms , then there exist holomorphic functions ''f''ij and such that, on a possibly smaller domain, :\omega^j = \sum_^r f_i^jdg^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 is not restrictive.


Higher degree forms

The statement does not generalize to higher degree forms, although there is a number of partial results such as
Darboux's theorem Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among ...
and the Cartan-Kähler theorem.


History

Despite being named for
Ferdinand Georg Frobenius Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous ...
, the theorem was first proven by
Alfred Clebsch Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. ...
and
Feodor Deahna Heinrich Wilhelm Feodor Deahna (8 July 1815 – 8 January 1844) was a German mathematician.. He is known for providing proof of what is now known as Frobenius theorem (differential topology), Frobenius theorem in differential topology, which he pub ...
. Deahna was the first to establish the
sufficient In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
conditions for the theorem, and Clebsch developed the necessary conditions. Frobenius is responsible for applying the theorem to
Pfaffian system In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...
s, thus paving the way for its usage in differential topology.


Applications

* In
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, the integrability of a system's constraint equations determines whether the system is holonomic or nonholonomic.


See also

*
Integrability conditions for differential systems In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...
*
Domain-straightening theorem In differential calculus, the domain-straightening theorem states that, given a vector field X on a manifold, there exist local coordinates y_1, \dots, y_n such that X = \partial / \partial y_1 in a neighborhood of a point where X is nonzero. 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 compl ...


Notes


References

* * * * * {{refend Theorems in differential geometry Theorems in differential topology Differential systems Foliations