Frobenius theorem (differential topology)
Encyclopedia
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 equation
s. 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 field
s (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
.
s. Suppose that fki(x) are a collection of real-valued C1 functions on Rn, for i = 1, 2, ..., n, and k = 1, 2, ..., r, where r < n, such that the matrix (fki) has rank r. Consider the following system of partial differential equations for a real-valued C2 function u on Rn:
(1)
One seeks conditions on the existence of a collection of solutions u1, ..., un−r such that the gradients
are linearly independent.
The Frobenius theorem asserts that this problem admits a solution locally if, and only if, the operators Lk satisfy a certain integrability condition known as involutivity. Specifically, they must satisfy relations of the form
for i, j = 1, 2,..., r, and all C2 functions u, and for some coefficients ckij(x) that are allowed to depend on x.
In other words, the commutator
s [Li,Lj] must lie in the linear span
of the Lk 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 Li so that the resulting operators do commute, and then to show that there is a coordinate system
yi for which these are precisely the partial derivatives with respect to y1, ..., yr.
clearly lacks a unique solution. Nevertheless, the 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 R3 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:
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 u1,...,un−r are solutions of the problem (1) satisfying the independence condition on the gradients. Consider the level set
s of (u1,...,un-r) regarded as an Rn−r-valued function. If v1,...,vn−r is any other 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 as the us, 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.
s. An alternative formulation, which is somewhat more intuitive, uses vector field
s.
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 can be integrated to define a family of curve
s. The integrability follows because the equation defining the curve is a first-order ordinary differential equation
, and thus its integrability is guaranteed by the Picard–Lindelöf theorem. Indeed, vector fields are often defined to be the derivatives of a collection of smooth curves.
This idea of integrability can be extended to collections of vector fields as well. One says that a subbundle
of the tangent bundle
TM is integrable (or involutive), if, for any two vector fields X and Y taking values in E, then the Lie bracket
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.
A subbundle
may also be defined to arise from a foliation
of a manifold. Let
be a submanifold that is a leaf of a foliation. Consider the tangent bundle TN. If TN is exactly E restricted to N, then one says that E arises from a regular foliation of M. Again, this definition is purely local: the foliation is defined only on charts.
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.
r, the rank being constant in value over U. The Frobenius theorem states that F is integrable
if and only if for every
the stalk Fp 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 form
s and Lie derivative
s. Frobenius' theorem is one of the basic tools for the study of vector field
s and foliations.
There are thus two forms of the theorem: one which operates with distribution
s, 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 α ∈ Ω(M) such that
for all v ∈ D, where i denotes the interior product of a vector field with a k-form. 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.
s, and A ⊂ X, B ⊂ Y a pair of open set
s. Let
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)
if u′(x) = F(x,u(x)) for all x ∈ A.
The equation (1) is completely integrable if for each
, 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 groundfield 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×B if and only if
for all s1, s2 ∈ X. Here the dot product denotes the action of the linear operator F(x,y) ∈ L(X,Y), as well as the action of the operator D1(x,y) ∈ L(X,L(X,Y)) and D2(x,y) ∈ L(Y, L(X,Y)).
s. 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 ∈ 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 Ep:
On the other hand, E is integrable if, for each p ∈ M, there is an immersed submanifold φ : N → M whose image contains p, such that the differential of φ is an isomorphism of TN with φ-1E.
The Frobenius theorem states that a subbundle E is integrable if and only if it is involutive.
s — manifolds over C with biholomorphic transition function
s.
Specifically, if
are r linearly independent holomorphic 1-forms on an open set in Cn such that
for some system of holomorphic 1-forms ψij, i,j=1,...,r, then there exist holomorphic functions fij and gi such that, on a possibly smaller domain,
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 Cn is not restrictive.
and the Cartan-Kähler theorem.
, 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.
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...
, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system
Overdetermined system
In mathematics, a system of linear equations is considered overdetermined if there are more equations than unknowns. The terminology can be described in terms of the concept of counting constraints. Each unknown can be seen as an available degree of freedom...
of first-order homogeneous linear 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. In modern geometric terms, the theorem gives necessary and sufficient conditions for the existence of 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 each of whose tangent bundles are spanned by a given family of vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
s (satisfying an integrability condition) in much the same way as 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...
may be assigned to a single vector field. The theorem is foundational in differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
and calculus on manifolds
Calculus on Manifolds
Michael Spivak's Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus is a short text treating analysis in several variables in Euclidean spaces and on differentiable 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 equationPartial 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. Suppose that fki(x) are a collection of real-valued C1 functions on Rn, for i = 1, 2, ..., n, and k = 1, 2, ..., r, where r < n, such that the matrix (fki) has rank r. Consider the following system of partial differential equations for a real-valued C2 function u on Rn:
(1)One seeks conditions on the existence of a collection of solutions u1, ..., un−r such that the gradients
are linearly independent.
The Frobenius theorem asserts that this problem admits a solution locally if, and only if, the operators Lk satisfy a certain integrability condition known as involutivity. Specifically, they must satisfy relations of the form
for i, j = 1, 2,..., r, and all C2 functions u, and for some coefficients ckij(x) that are allowed to depend on x.
In other words, the commutator
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:...
s [Li,Lj] must lie in the linear span
Linear span
In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...
of the Lk 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 Li so that the resulting operators do commute, and then to show that there is a coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
yi for which these are precisely the partial derivatives with respect to y1, ..., yr.
From analysis to geometry
Solutions to underdetermined systems of equations are seldom unique. For example, the systemclearly lacks a unique solution. Nevertheless, the 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 R3 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:
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 u1,...,un−r are solutions of the problem (1) satisfying the independence condition on the gradients. Consider the level set
Level set
In mathematics, a level set of a real-valued function f of n variables is a set of the formthat is, a set where the function takes on a given constant value c....
s of (u1,...,un-r) regarded as an Rn−r-valued function. If v1,...,vn−r is any other such collection of solutions, one can show (using some linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
and the mean value theorem
Mean value theorem
In calculus, the mean value theorem states, roughly, that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the...
) that this has the same family of level sets as the us, 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 systems, which today can be translated into the language of differential formDifferential 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. An alternative formulation, which is somewhat more intuitive, uses vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
s.
Formulation using vector fields
In the vector field formulation, the theorem states that a subbundleSubbundle
In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces Ux of the fibers Vx of V at x in X, that make up a vector bundle in their own right....
of the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
of a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
is integrable (or involutive) if and only if it arises from a regular foliation. In this context, the Frobenius theorem relates integrability
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 way a differential form restricts to a...
to foliation; to state the theorem, both concepts must be clearly defined.
One begins by noting that an arbitrary smooth vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
X on a manifold M can be integrated to define a family of curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
s. The integrability follows because the equation defining the curve is a first-order ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
, and thus its integrability is guaranteed by the Picard–Lindelöf theorem. Indeed, vector fields are often defined to be the derivatives of a collection of smooth curves.
This idea of integrability can be extended to collections of vector fields as well. One says that a subbundle
Subbundle
In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces Ux of the fibers Vx of V at x in X, that make up a vector bundle in their own right....
of the tangent bundleTangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
TM is integrable (or involutive), if, for any two vector fields X and Y taking values in E, then the Lie bracket
Lie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
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.A subbundle
may also be defined to arise from a foliationFoliation
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....
of a manifold. Let
be a submanifold that is a leaf of a foliation. Consider the tangent bundle TN. If TN is exactly E restricted to N, then one says that E arises from a regular foliation of M. Again, this definition is purely local: the foliation is defined only on charts.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, Ω1(U) be the space of smooth, differentiable 1-forms on U, and F be a submodule of Ω1(U) of rankRank (linear algebra)
The column rank of a matrix A is the maximum number of linearly independent column vectors of A. The row rank of a matrix A is the maximum number of linearly independent row vectors of A...
r, the rank being constant in value over U. The Frobenius theorem states that F is integrable
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 way a differential form restricts to a...
if and only if for every
the stalk Fp 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
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....
. The correspondence to the definition in terms of vector fields given in the introduction follows from the close relationship between 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 and Lie derivative
Lie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
s. Frobenius' theorem is one of the basic tools for the study of vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
s and foliations.
There are thus two forms of the theorem: one which operates with distribution
Distribution (differential geometry)
In differential geometry, a discipline within mathematics, a distribution is a subset of the tangent bundle of a manifold satisfying certain properties...
s, 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 α ∈ Ω(M) such that
for all v ∈ D, where i denotes the interior product of a vector field with a k-form. The set I(D) forms a subring and, in fact, an ideal in Ω(M). Furthermore, using the definition of 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....
, it can be shown that I(D) is closed under exterior differentiation (it is a differential ideal
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 exterior differentiation d...
) 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.
Infinite dimensions
One infinite-dimensional generalization is as follows. Let X and Y be Banach spaceBanach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s, and A ⊂ X, B ⊂ Y a pair of open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
s. Let
be a continuously differentiable function of the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
(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)if u′(x) = F(x,u(x)) for all x ∈ A.
The equation (1) is completely integrable if for each
, 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 groundfield 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×B if and only if
for all s1, s2 ∈ X. Here the dot product denotes the action of the linear operator F(x,y) ∈ L(X,Y), as well as the action of the operator D1(x,y) ∈ L(X,L(X,Y)) and D2(x,y) ∈ L(Y, L(X,Y)).
Banach manifolds
The infinite dimensional version of the Frobenius theorem also holds on Banach manifoldBanach 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...
s. 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 ∈ 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 Ep:
On the other hand, E is integrable if, for each p ∈ M, there is an immersed submanifold φ : N → M whose image contains p, such that the differential of φ is an isomorphism of TN with φ-1E.
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 manifoldComplex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
s — manifolds over C with biholomorphic transition function
Transition function
In mathematics, a transition function has several different meanings:* In topology, a transition function is a homeomorphism from one coordinate chart to another...
s.
Specifically, if
are r linearly independent holomorphic 1-forms on an open set in Cn such thatfor some system of holomorphic 1-forms ψij, i,j=1,...,r, then there exist holomorphic functions fij and gi such that, on a possibly smaller domain,
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 Cn 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 theoremDarboux'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 them being symplectic geometry...
and the Cartan-Kähler theorem.
History
Despite being named for Ferdinand Georg FrobeniusFerdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
, the theorem was first proven by Alfred Clebsch
Alfred Clebsch
Rudolf Friedrich Alfred Clebsch 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. He subsequently taught in Berlin and Karlsruhe...
and Feodor Deahna
Feodor Deahna
Heinrich Wilhelm Feodor Deahna was a German mathematician, best known for providing proof of what is now known as Frobenius theorem in 1840.- Selected works :...
. 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.
See also
- Integrability conditions for differential systemsIntegrability conditions for differential systemsIn 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 way a differential form restricts to a...
- Domain-straightening theoremDomain-straightening theoremIn 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...