Theorem of Frobenius (differential geometry) – Wikipedia

Posted on March 28, 2019 by lordneo

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The Frobenius theorem gives a necessary and sufficient condition for local integrability of a system of partial derivative equations of the first order whose right -wing member depends on the variables, the unknowns, but does not depend on partial derivatives of these unknowns: such a system D ‘Partial derivative equations is called a “PFAFF system”. The functions of the second member are assumed only to class

{displaystyle C^{1}}

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$C^{1}$ , which makes it impossible to apply the Cauchy-Kowalevski theorem, which supposes these analytical functions. The Frobenius theorem has close links with the Poincaré lemma for 1-formas, this lemma then indicating under what condition a differential 1-form

{displaystyle omega }

$omega$ is locally exact. The Theorem of Frobenius leads to consider the “integral varieties” of differential geometry and can express themselves in this language. These full varieties lead to the concept of puff pastry ^{[ first ]}. The “Frobenius theorem” was actually established by Feodor Deahna (in) In 1840, in an article deepening the work of Johann Friedrich Pfaff and Charles Gustave Jacob Jacobi on the partial derivative equations of the first order (going back to 1815 and 1827 respectively) and which went unnoticed until Ferdinand Georg Frobenius exhumes it in 1875 ^{[ 2 ]}. The theorem of Chow-Rashevskii (in) and that of Hector Sussmann, dating from 1938-39 and 1973 respectively ^{[ 3 ]}^,^{[ 4 ]}, study the existence of full varieties for ” p -Champs »singular; They are, like the theorem of Frobenius, widely used to study the commandability of non -linear systems ^{[ 5 ]}(the link between this question of commandability and the Theorem of Frobenius was first noted by Robert Hermann (in) in 1963).

Table of Contents

Frobenius theorem: “functional” formulation [ modifier | Modifier and code ]

Either IN an open of

{displaystyle mathbb {R} ^{p}}

${displaystyle mathbb {R} ^{p}}$ , IN an open of

{displaystyle mathbb {R} ^{n-p}}

${displaystyle mathbb {R} ^{n-p}}$ , and, for everything k ,

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{Displaystyle 1Leq Kleq {n-p}}

${displaystyle 1leq kleq {n-p}}$ , a function

{displaystyle B^{k}:Utimes Vrightarrow mathbb {R} ^{p}}

${displaystyle B^{k}:Utimes Vrightarrow mathbb {R} ^{p}}$ classy

{displaystyle C^{r}}

$C^{r}$ (

{displaystyle 1leq rleq +infty }

${displaystyle 1leq rleq +infty }$ ). Consider the system (f) of partial derivative equations, or “pfaff system”

(F) :

{displaystyle {frac {partial v^{k}}{partial x^{h}}}=B_{h}^{k}left(x^{1},…,x^{p},v^{1},…,v^{n-p}right)quad left(1leq kleq n-p,1leq hleq pright).}

A integral variety of this system, if it exists, is a sub-variety of N of

{displaystyle Utimes V}

${displaystyle Utimes V}$ , of class

{displaystyle C^{r}}

${displaystyle C^{r}}$ , defined by parametric representation (RP):

(Rp):

{displaystyle x^{p+k}=v^{k}left(x^{1},…,x^{p}right)}

On which the differential (or “pfaff forms”) linearly independently independently arranged.

{displaystyle omega ^{k}=dx^{p+k}-sum limits _{h=1}^{p}{frac {partial v^{k}}{partial x^{h}}}dx^{h}}

Resolve the PFAFF (F) system is equivalent to determining an integral variety N From this system, and (f) admit a solution if, and only if such an integral variety exists.

Frobenius theorem in functional form – The following conditions are equivalent: (1) for any point

{displaystyle (x_{0},v_{0})in Utimes V}

${displaystyle (x_{0},v_{0})in Utimes V}$ There is an open neighborhood

{displaystyle Ssubset U}

${displaystyle Ssubset U}$ of

{displaystyle x_{0}}

$x_{0}$ , an open neighborhood

{displaystyle Tsubset V}

${displaystyle Tsubset V}$ of

{displaystyle v_{0}}

$v_0$ , and a single function in classy

{displaystyle C^{r}}

$C^{r}$ , of S In T , solution of (f) and such that

{displaystyle v(x_{0})=v_{0}}

${displaystyle v(x_{0})=v_{0}}$ . (2) Functions

{displaystyle B^{k}}

${displaystyle B^{k}}$ check in

{displaystyle Utimes V}

${displaystyle Utimes V}$ The “Frobenius integrability condition”

{displaystyle {frac {partial B_{h}^{k}}{partial x^{l}}}+sum _{j=1}^{n-p}{frac {partial B_{h}^{k}}{partial v^{j}}}B_{l}^{j}={frac {partial B_{l}^{k}}{partial x^{h}}}+sum _{j=1}^{n-p}{frac {partial B_{l}^{k}}{partial v^{j}}}B_{h}^{j}}

{displaystyle left(1leq kleq n-p,1leq lleq p,1leq hleq pright).}

Demonstration

The condition is necessary, because if there is such a function in , she is necessarily class

{displaystyle C^{r+1}}

${displaystyle C^{r+1}}$ (with the Convention that

{displaystyle +infty +1=+infty }

${displaystyle +infty +1=+infty }$ ). So, according to the Schwarz theorem,

{displaystyle {frac {partial ^{2}v^{k}}{partial x^{h}partial x^{l}}}={frac {partial ^{2}v^{k}}{partial x^{l}partial x^{h}}}}

${displaystyle {frac {partial ^{2}v^{k}}{partial x^{h}partial x^{l}}}={frac {partial ^{2}v^{k}}{partial x^{l}partial x^{h}}}}$ .

Or,

{displaystyle {frac {partial ^{2}v^{k}}{partial x^{h}partial x^{l}}}={frac {partial B_{h}^{k}}{partial x^{l}}}+sum _{j=1}^{n-p}{frac {partial B_{h}^{k}}{partial x^{j+p}}}{frac {partial v^{j}}{partial x^{l}}}={frac {partial B_{h}^{k}}{partial x^{l}}}+sum _{j=1}^{n-p}{frac {partial B_{h}^{k}}{partial x^{j+p}}}B_{l}^{j}}

${displaystyle {frac {partial ^{2}v^{k}}{partial x^{h}partial x^{l}}}={frac {partial B_{h}^{k}}{partial x^{l}}}+sum _{j=1}^{n-p}{frac {partial B_{h}^{k}}{partial x^{j+p}}}{frac {partial v^{j}}{partial x^{l}}}={frac {partial B_{h}^{k}}{partial x^{l}}}+sum _{j=1}^{n-p}{frac {partial B_{h}^{k}}{partial x^{j+p}}}B_{l}^{j}}$ , and the condition of Frobenius results from it.

Conversely, the Frobenius integrability condition is equivalent to

{displaystyle domega ^{k}=0}

${displaystyle domega ^{k}=0}$ with

{displaystyle omega ^{k}left(x^{1},…,x^{n}right)=dx^{p+k}-sum limits _{h=1}^{p}B_{h}^{k}left(x^{1},…,x^{n}right)dx^{h}}

Poincaré’s lemma then implies the existence of functions

{displaystyle f^{k}}

${displaystyle f^{k}}$ , defined in an open neighboring contractile sufficiently small IN of

{displaystyle left(x_{0},v_{0}right)}

${displaystyle left(x_{0},v_{0}right)}$ , as

{displaystyle omega ^{k}=df^{k}left(1leq kleq n-pright)}

${displaystyle omega ^{k}=df^{k}left(1leq kleq n-pright)}$ . The variety

{displaystyle Ncap W}

${displaystyle Ncap W}$ is therefore defined by the equations

(No) ::

{displaystyle f^{k}left(x^{1},…,x^{p},x^{p+1},…,x^{n}right)=f^{k}left(x_{0}^{1},…,x_{0}^{p},v_{0}^{1},…,v_{0}^{n-p}right).}

${displaystyle f^{k}left(x^{1},...,x^{p},x^{p+1},...,x^{n}right)=f^{k}left(x_{0}^{1},...,x_{0}^{p},v_{0}^{1},...,v_{0}^{n-p}right).}$

Since

{displaystyle df^{k}}

${displaystyle df^{k}}$ are linear independently, the theorem of implicit functions implies the existence of a neighborhood

{displaystyle Stimes Tsubset W}

${displaystyle Stimes Tsubset W}$ of

{displaystyle left(x_{0},v_{0}right)}

${displaystyle left(x_{0},v_{0}right)}$ and functions

{displaystyle v^{k}left(1leq kleq n-pright)}

${displaystyle v^{k}left(1leq kleq n-pright)}$ classy

{displaystyle C^{r}}

${displaystyle C^{r}}$ , defined in a unique way, such as equations (IS) equivalent to (RP) in

{displaystyle Stimes T}

${displaystyle Stimes T}$ ; These functions check

{displaystyle v(x_{0})=v_{0}}

${displaystyle v(x_{0})=v_{0}}$ .

Noticed [ modifier | Modifier and code ]

There is a generalization of this theorem in case

{displaystyle mathbb {R} ^{p}}

${displaystyle mathbb {R} ^{p}}$ And

{displaystyle mathbb {R} ^{n-p}}

${displaystyle mathbb {R} ^{n-p}}$ are replaced by Banach spaces ^{[ 6 ]}.

From now on,

{displaystyle r=+infty }

$r=+infty$ and all differential varieties (which will simply be called varieties ) are class

{DisplayStyle C^{Infty}}

$C^{infty }$ . Either M A variety of dimension n . We designate by

{displaystyle {mathcal {E}}_{0}left(Mright)}

${displaystyle {mathcal {E}}_{0}left(Mright)}$ the

{displaystyle mathbb {R} }

$mathbb {R}$ -Algèbre of indefinitely derivable functions on variety M and by

{displaystyle {mathcal {T}}_{0}^{1}left(Mright)}

${displaystyle {mathcal {T}}_{0}^{1}left(Mright)}$ the

{displaystyle {mathcal {E}}_{0}left(Mright)}

${displaystyle {mathcal {E}}_{0}left(Mright)}$ -Modle of class vector fields

{DisplayStyle C^{Infty}}

$C^{infty }$ on M . By definition,

{displaystyle {mathcal {T}}_{0}^{1}left(Mright)}

${displaystyle {mathcal {T}}_{0}^{1}left(Mright)}$ is the set of the fiber -tangent fiber

{displaystyle Tleft(Mright)}

${displaystyle Tleft(Mright)}$ .

Either ${displaystyle fin {mathcal {E}}_{0}left(Mright),Xin {mathcal {T}}_{0}^{1}left(Mright)}$
Given ${displaystyle X,Yin {mathcal {T}}_{0}^{1}left(Mright)}$

Lie hook is an application ${displaystyle mathbb {R} }$

{displaystyle Z^{i}=sum limits _{j=1}^{n}left({frac {partial Y^{i}}{partial xi ^{j}}}X^{j}-{frac {partial X^{i}}{partial xi ^{j}}}Y^{j}right)}

The LIE hook has the following “land ownership”: either M , N two varieties, ${displaystyle varphi :Mrightarrow N}$
Either ${displaystyle {mathcal {E}}_{p}left(Mright)=Omega ^{p}left(Mright)}$

{displaystyle leftlangle domega ,Xwedge Yrightrangle ={mathcal {L}}_{X}.leftlangle omega ,Yrightrangle -{mathcal {L}}_{Y}.leftlangle omega ,Xrightrangle -leftlangle omega ,left[X,Yright]rightrangle }

Either ${displaystyle f_{1},f_{2}in {mathcal {E}}_{0}left(Mright),X_{1},X_{2}in {mathcal {T}}_{0}^{1}left(Mright)}$

{displaystyle left[f_{1}X_{1},f_{2}X_{2}right]=f_{1}f_{2}left[X_{1},X_{2}right]+left(f_{1}{mathcal {L}}_{X_{1}}.f_{2}right)X_{2}-left(f_{2}{mathcal {L}}_{X_{2}}.f_{1}right)X_{1}.}

So we have the following result:

Lemme – And

{displaystyle left[X_{1},X_{2}right]=0}

${displaystyle left[X_{1},X_{2}right]=0}$ , then whatever the vector fields

{displaystyle Y,Z}

${displaystyle Y,Z}$ belonging to

{displaystyle {mathcal {E}}_{0}left(Mright)}

${displaystyle {mathcal {E}}_{0}left(Mright)}$ -module

{displaystyle spanleft{X_{1},X_{2}right}}

${displaystyle spanleft{X_{1},X_{2}right}}$ generated by

{displaystyle X_{1}}

$X_{{1}}$ And

{displaystyle X_{2}}

$X_{{2}}$ , the lie crochet

{displaystyle left[Y,Zright]}

${displaystyle left[Y,Zright]}$ belongs to

{displaystyle spanleft{X_{1},X_{2}right}}

${displaystyle spanleft{X_{1},X_{2}right}}$ .

Example [ modifier | Modifier and code ]

Consider the elementary case where

{displaystyle p=2,n=3}

${displaystyle p=2,n=3}$ And let’s see how the theorem of Frobenius in its functional form is expressed in the geometric formalism of the Lie hooks, by bringing itself to the situation where M is an open of

{displaystyle mathbb {R} ^{3}}

${mathbb R}^{3}$ . Let’s ask

{displaystyle Delta =spanleft{X_{1},X_{2}right}}

${displaystyle Delta =spanleft{X_{1},X_{2}right}}$ with

{displaystyle X_{1}=left(1,0,{frac {partial v}{partial x^{1}}}right)=left(1,0,B_{1}right)}

{displaystyle X_{2}=left(0,1,{frac {partial v}{partial x^{2}}}right)=left(0,1,B_{2}right)}

Frobenius’ integrability condition is written, with

{displaystyle B_{h}^{1}=B_{h}=B_{h}left(x_{1},x_{2},x_{3}right)}

${displaystyle B_{h}^{1}=B_{h}=B_{h}left(x_{1},x_{2},x_{3}right)}$ ,

{displaystyle {frac {partial B_{1}}{partial x^{2}}}+{frac {partial B_{1}}{partial x^{3}}}B_{2}={frac {partial B_{2}}{partial x^{1}}}+{frac {partial B_{2}}{partial x^{3}}}B_{1}}

equivalent to

{displaystyle left[X_{1},X_{2}right]=0}

${displaystyle left[X_{1},X_{2}right]=0}$ . Consequently, Frobenius’s fulfillability condition leads, according to the lemma above, only for all fields of vectors

{displaystyle Y,Zin Delta }

${displaystyle Y,Zin Delta }$ ,on a

{displaystyle left[Y,Zright]in Delta }

${displaystyle left[Y,Zright]in Delta }$ . As we will see further, we can express this by saying that the “2-champ”

{displaystyle Delta }

$Delta$ is “involved”.

The recovery theorem of landmarks generalizes the theorem for recovery of vector fields.

Recovery theorem of landmarks – Either M A variety of dimension n ,

{displaystyle x_{0}}

$x_{0}$ a point of M And

{displaystyle X_{1},…,X_{p}}

${displaystyle X_{1},...,X_{p}}$ vector fields on M such as

{displaystyle X_{1}left(x_{0}right),…,X_{p}left(x_{0}right)}

${displaystyle X_{1}left(x_{0}right),...,X_{p}left(x_{0}right)}$ are linear independent. The following conditions are equivalent: (1) LIE hooks

{displaystyle left[X_{i},X_{j}right]}

${displaystyle left[X_{i},X_{j}right]}$ are all zero

{displaystyle (i,j=1,…,p)}

${displaystyle (i,j=1,...,p)}$ . (2) There is a card

{displaystyle left(U,varphi ,nright)}

${displaystyle left(U,varphi ,nright)}$ centered on

{displaystyle x_{0}}

$x_{0}$ such as

{displaystyle varphi _{ast }left(X_{i}leftvert _{U}right.right)={frac {partial }{partial x^{i}}}.}

${displaystyle varphi _{ast }left(X_{i}leftvert _{U}right.right)={frac {partial }{partial x^{i}}}.}$

(The question being local, we can assume that

{displaystyle M=U}

${displaystyle M=U}$ is an open of

{displaystyle mathbb {R} ^{n}}

$mathbb {R} ^{n}$ . The condition is necessary, because the land of the lie hute implies

{displaystyle varphi _{ast }left(left[X_{i},X_{j}right]right)=left[varphi _{ast }left(X_{i}right),varphi _{ast }left(X_{j}right)right]=left[{frac {partial }{partial x^{i}}},{frac {partial }{partial x^{i}}}right]=0}

${displaystyle varphi _{ast }left(left[X_{i},X_{j}right]right)=left[varphi _{ast }left(X_{i}right),varphi _{ast }left(X_{j}right)right]=left[{frac {partial }{partial x^{i}}},{frac {partial }{partial x^{i}}}right]=0}$ . We show that it is sufficient thanks to the theory of differential equations ^{[ 7 ]}.)

Frobenius theorem: geometric formulation [ modifier | Modifier and code ]

Let’s start with some definitions.

(1) and p -champ (or a p -Direction, or a distribution of dimension contact elements p , or a dimension subfibrum p Fiber tangent

{displaystyle Tleft(Mright)}

${displaystyle Tleft(Mright)}$ ) class

{DisplayStyle C^{Infty}}

$C^{infty }$ is an application

{displaystyle Delta :Mni xmapsto Delta _{x}}

${displaystyle Delta :Mni xmapsto Delta _{x}}$ Or

{displaystyle Delta _{x}}

${displaystyle Delta _{x}}$ is a dimension subspace p Tangent space

{displaystyle T_{x}left(Mright)}

${displaystyle T_{x}left(Mright)}$ To M on point x , checking the following condition: for everything

{Displaystyle please

$xin M$ , there is an open neighborhood IN of x In M and vector fields

{displaystyle X_{1},…,X_{p}in {mathcal {T}}_{0}^{1}left(Mright)}

${displaystyle X_{1},...,X_{p}in {mathcal {T}}_{0}^{1}left(Mright)}$ such as

{displaystyle X_{1}(y),…,X_{p}(y)}

${displaystyle X_{1}(y),...,X_{p}(y)}$ form a base of

{Displaystyle Delta _ {y}}

${displaystyle Delta _{y}}$ for everything

{displaystyle yin U}

${displaystyle yin U}$ (We write then

{Displaystyle Delta _ {y} = spanleft {x_ {1} (y), …, x_ {p} (y) right}}}}

${displaystyle Delta _{y}=spanleft{X_{1}(y),...,X_{p}(y)right}}$ And

{displaystyle Delta =spanleft{X_{1},…,X_{p}right}}

${displaystyle Delta =spanleft{X_{1},...,X_{p}right}}$ , this last writing meaning that

{displaystyle Delta }

$Delta$ is the

{displaystyle {mathcal {E}}_{0}left(Uright)}

${displaystyle {mathcal {E}}_{0}left(Uright)}$ -Module engendered by

{displaystyle X_{1},…,X_{p}}

${displaystyle X_{1},...,X_{p}}$ ). In the following, ” p -champ “means” p -Champ

{DisplayStyle C^{Infty}}

$C^{infty }$ ».

(2) a sub-variety N of M is called a integral variety of p -champ

{displaystyle Delta }

$Delta$ If for everything

{Displaystyle please

$xin M$ , and by designating by

{displaystyle iota :Nrightarrow M}

${displaystyle iota :Nrightarrow M}$ l’inclusion,

{displaystyle iota _{ast }left(T_{x}left(Nright)right)=Delta _{x}}

${displaystyle iota _{ast }left(T_{x}left(Nright)right)=Delta _{x}}$ (In other words, the tangent space

{displaystyle T_{x}left(Nright)}

${displaystyle T_{x}left(Nright)}$ identify with the subspace

{displaystyle Delta _{x}subset T_{x}left(Mright)}

${displaystyle Delta _{x}subset T_{x}left(Mright)}$ ). This full variety is said maximum If any integral variety that contains it coincides with it (it is then dimension p ^{[ 8 ]}). The notion of integration is local and invariant by differential.

(3) the p -champ

{displaystyle Delta }

$Delta$ is said completely integrable If he admits an integral variety. It is said engaging and

{displaystyle left[X_{1},X_{2}right]in Delta }

${displaystyle left[X_{1},X_{2}right]in Delta }$ for everyone

{displaystyle X_{1},X_{2}in Delta }

${displaystyle X_{1},X_{2}in Delta }$ .

(4) for everything

{Displaystyle please

$x in M$ , either

{displaystyle Delta _{x}^{0}}

${displaystyle Delta _{x}^{0}}$ The polar of

{displaystyle Delta _{x}}

${displaystyle Delta _{x}}$ , that is to say the subspace of the cotangent space

{displaystyle T_{x}^{ast }left(Mright)}

${displaystyle T_{x}^{ast }left(Mright)}$ Orthogonal at

{displaystyle Delta _{x}}

${displaystyle Delta _{x}}$ , And

{displaystyle omega ^{j}left(xright)}

${displaystyle omega ^{j}left(xright)}$ A base of

{displaystyle Delta _{x}^{0}}

${displaystyle Delta _{x}^{0}}$ . L’application

{displaystyle Delta ^{0}:xmapsto Delta _{x}^{0}}

${displaystyle Delta ^{0}:xmapsto Delta _{x}^{0}}$ , if she’s class

{DisplayStyle C^{Infty}}

$C^{infty }$ (notion that we define by “dualisant” that of p -Champ

{DisplayStyle C^{Infty}}

$C^{infty }$ ), is a codistribution , namely a

{displaystyle {mathcal {E}}_{0}left(Mright)}

${displaystyle {mathcal {E}}_{0}left(Mright)}$ -module, having as a basis

{displaystyle n-p}

${displaystyle n-p}$ 1-form (or forms of PFAFF)

{displaystyle omega ^{j}left(1leq jleq n-pright)}

${displaystyle omega ^{j}left(1leq jleq n-pright)}$ . These forms of pfaff are canceled on N , namely that for any field of vectors

{displaystyle Xin {mathcal {T}}_{0}^{1}left(Nright)}

${displaystyle Xin {mathcal {T}}_{0}^{1}left(Nright)}$ ,

{displaystyle leftlangle omega ^{j},Xrightrangle =0left(1leq jleq n-pright)}

${displaystyle leftlangle omega ^{j},Xrightrangle =0left(1leq jleq n-pright)}$ . It is also said that the PFAFF system

(P)::

{displaystyle omega ^{j}=0left(1leq jleq n-pright)}

${displaystyle omega ^{j}=0left(1leq jleq n-pright)}$

where the

{displaystyle omega ^{j}}

${displaystyle omega ^{j}}$ are linear independently, is associated At p -champ

{displaystyle Delta }

$Delta$ and defines the integral variety N .

(5) either

{displaystyle Omega ^{q}left(Mright)}

${displaystyle Omega ^{q}left(Mright)}$ The vector space of degree forms q on M And

{displaystyle Omega left(Mright)}

${displaystyle Omega left(Mright)}$ graduated algebra defined by

{displaystyle Omega left(Mright)=bigoplus limits _{q=0}^{+infty }Omega ^{q}left(Mright)}

We designate by

{displaystyle {mathfrak {O}}left(Delta right)}

${displaystyle {mathfrak {O}}left(Delta right)}$ the ideal graduated of

{displaystyle Omega left(Mright)}

${displaystyle Omega left(Mright)}$ constituted by forms

{displaystyle omega }

$omega$ checking the following condition: for any q -form

{displaystyle omega in {mathfrak {O}}left(Delta right)}

${displaystyle omega in {mathfrak {O}}left(Delta right)}$ And all vector fields

{displaystyle X_{1},…,X_{q}in Delta }

${displaystyle X_{1},...,X_{q}in Delta }$ ,

{displaystyle leftlangle omega ,X_{1}wedge …wedge X_{q}rightrangle =0}

Finally, we designate by

{displaystyle dleft({mathfrak {O}}left(Delta right)right)}

${displaystyle dleft({mathfrak {O}}left(Delta right)right)}$ the

{displaystyle mathbb {R} }

$mathbb {R}$ -vector spaces made up of

{displaystyle domega left(omega in {mathfrak {O}}left(Delta right)right)}

${displaystyle domega left(omega in {mathfrak {O}}left(Delta right)right)}$ .

Frobenius theorem in geometric form – Either

{displaystyle Delta }

$Delta$ and p -champ on a variety M and (p) the associated pfaff system. The following conditions are equivalent:

(i)

{displaystyle Delta }

(ii)

{displaystyle Delta }

(iii)

{displaystyle dleft({mathfrak {O}}left(Delta right)right)subset {mathfrak {O}}left(Delta right)}

(iv) for everything

{Displaystyle please

{displaystyle domega ^{j}=sum limits _{k=1}^{n-p}omega ^{k}wedge alpha _{k}^{j}quad left(1leq jleq n-pright)}

(V) for everything

{Displaystyle please

{displaystyle omega ^{j}=sum limits _{k}f_{k}^{j}dg^{k}}

Demonstration

(i) ⇒ (ii) by the theorem of Frobenius in functional form and the lemma.

(ii) ⇒ (i): suppose

{displaystyle left[Y,Zright]}

${displaystyle left[Y,Zright]}$ for everyone

{displaystyle X,Yin Delta }

${displaystyle X,Yin Delta }$ . Suppose moreover, without loss of generality (the condition (i) being local), that M Or an open of

{displaystyle mathbb {R} ^{n}}

$mathbb {R} ^{n}$ and

{displaystyle Delta _{0}}

$Delta _{0}$ either generated by p first vectors

{displaystyle mathbf {e} _{i}={frac {partial }{partial x^{i}}}left(0right)}

${displaystyle mathbf {e} _{i}={frac {partial }{partial x^{i}}}left(0right)}$ of the canonical base of

{displaystyle mathbb {R} ^{n}}

$mathbb {R} ^{n}$ . Either

{displaystyle pi :mathbb {R} ^{n}rightarrow mathbb {R} ^{p}}

${displaystyle pi :mathbb {R} ^{n}rightarrow mathbb {R} ^{p}}$ Canonical projection. The application

{displaystyle pi _{ast }leftvert _{Delta _{x}}right.}

${displaystyle pi _{ast }leftvert _{Delta _{x}}right.}$ is an isomorphism of

{displaystyle Delta _{x}}

${displaystyle Delta _{x}}$ on

{displaystyle T_{x}left(mathbb {R} ^{p}right)cong mathbb {R} ^{p}}

${displaystyle T_{x}left(mathbb {R} ^{p}right)cong mathbb {R} ^{p}}$ for everything x in a neighborhood of 0 in M (neighborhood that can be assumed to be equal to M ). So we can find in M Vector fields

{displaystyle X_{1},…,X_{p}}

${displaystyle X_{1},...,X_{p}}$ such as

{displaystyle X_{1}left(xright),…,X_{p}left(xright)in Delta _{x}}

${displaystyle X_{1}left(xright),...,X_{p}left(xright)in Delta _{x}}$ And

{displaystyle pi _{ast }left(X_{i}right)={frac {partial }{partial x^{i}}}left(1leq ileq pright)}

${displaystyle pi _{ast }left(X_{i}right)={frac {partial }{partial x^{i}}}left(1leq ileq pright)}$ . Consequently, by fonctoriality of the LIE hook,

{displaystyle pi _{ast }left(left[X_{i},X_{j}right]_{x}right)=left[mathbf {e} _{i},mathbf {e} _{j}right]_{pi left(xright)}=0}

Since by hypothesis,

{displaystyle left[X_{i},X_{j}right]in Delta }

${displaystyle left[X_{i},X_{j}right]in Delta }$ , this leads

{displaystyle left[X_{i},X_{j}right]=0}

${displaystyle left[X_{i},X_{j}right]=0}$ . Consequently, according to the recovery theorem of landmarks, there is an open

{displaystyle Usubset M}

${displaystyle Usubset M}$ , which will be assumed to be equal to M , and a different

{displaystyle varphi :Mrightarrow N}

${displaystyle varphi :Mrightarrow N}$ , with

{displaystyle N=varphi (M)subset mathbb {R} ^{n}}

${displaystyle N=varphi (M)subset mathbb {R} ^{n}}$ , such as derivative

{displaystyle Dvarphi }

${displaystyle Dvarphi }$ send

{displaystyle X_{i}}

$X_{i}$ on

{displaystyle {frac {partial }{partial x^{i}}}left(1leq ileq pright)}

${displaystyle {frac {partial }{partial x^{i}}}left(1leq ileq pright)}$ . The variety N is integral, since defined by relations

{displaystyle x^{i}=C^{te}left(p+1leq ileq nright)}

${displaystyle x^{i}=C^{te}left(p+1leq ileq nright)}$ , and the variety M , which is different, is also integral. As a result,

{displaystyle Delta }

$Delta$ is completely integrable.

(i) ⇔ (iii): we only have to consider 1-form, for which the Maurer-Cartan formula applies. Either

{displaystyle omega in {mathfrak {O}}left(Delta right)}

${displaystyle omega in {mathfrak {O}}left(Delta right)}$ And

{displaystyle X,Yin Delta }

${displaystyle X,Yin Delta }$ . On a

{displaystyle leftlangle domega ,Xwedge Yrightrangle =0}

${displaystyle leftlangle domega ,Xwedge Yrightrangle =0}$ if and only if

{displaystyle leftlangle omega ,left[X,Yright]rightrangle =0}

${displaystyle leftlangle omega ,left[X,Yright]rightrangle =0}$ , that’s to say

{displaystyle left[X,Yright]in Delta }

${displaystyle left[X,Yright]in Delta }$ .

(iii) ⇔ (iv): external products

{displaystyle omega ^{i}wedge dx^{h}}

${displaystyle omega ^{i}wedge dx^{h}}$ ,

{displaystyle omega ^{i}wedge omega ^{k}}

${displaystyle omega ^{i}wedge omega ^{k}}$ ,

{displaystyle dx^{l}wedge dx^{h}}

${displaystyle dx^{l}wedge dx^{h}}$ ,

{Displaytlele Left (13 Lleq Kelq N-P,1LQ l

${displaystyle left(1leq jleq kleq n-p,1leq l<hleq pright)}$ form a base of

{displaystyle {mathcal {E}}_{0}left(Wright)}

${displaystyle {mathcal {E}}_{0}left(Wright)}$ -Module of 2-differentials on IN . So we can write for

{Displaystyle 1Leq N-P}

${displaystyle 1leq jleq n-p}$

{displaystyle domega ^{j}=sum limits _{k,h}C_{kh}^{j}omega ^{k}wedge dx^{h}+sum limits _{i,k}D_{ik}^{j}omega ^{i}wedge omega ^{k}+sum limits _{h,l}E_{hl}^{j}dx^{l}wedge dx^{h}}

where the coefficients belong to

{displaystyle {mathcal {E}}_{0}left(Wright)}

${displaystyle {mathcal {E}}_{0}left(Wright)}$ . Therefore, (iii) is verified if, and only if the

{displaystyle E_{hl}}

${displaystyle E_{hl}}$ are all zero, which is equivalent to (IV).

(i) ⇒ (v): Suppose (i) verified; We know that there is a card

{displaystyle left(U,xi ,nright)}

${displaystyle left(U,xi ,nright)}$ of M for which the integral sub-variety N is such that

{Displaystyle NCAP U}

${displaystyle Ncap U}$ for local contact details

{Displaystyle xi ^{p+1} = … = xi ^{n} = 0}

${displaystyle xi ^{p+1}=...=xi ^{n}=0}$ . 1-Form

{Displaystyle dxi ^{p+1}, …, dxi ^{n}}}

${displaystyle dxi ^{p+1},...,dxi ^{n}}$ therefore constitute a base of

{displaystyle {mathcal {E}}_{0}left(Uright)}

${displaystyle {mathcal {E}}_{0}left(Uright)}$ -module

{displaystyle Delta ^{0}}

${displaystyle Delta ^{0}}$ . As a result, 1-formmes

{displaystyle omega ^{j}left(1leq jleq n-pright)}

${displaystyle omega ^{j}left(1leq jleq n-pright)}$ are put in the form indicated in (V) with

{Displaystyle G ^{k} = xi ^{p+k}}}

${displaystyle g^{k}=xi ^{p+k}}$ .

(5) ⇒ (3): soit

{displaystyle omega in {mathfrak {O}}left(Delta right)}

${displaystyle omega in {mathfrak {O}}left(Delta right)}$ and suppose (v) verified where the 1-formas

{displaystyle dg^{k}}

${displaystyle dg^{k}}$ are linear independently (which can be assumed without loss of generality). Then vector fields

{displaystyle X_{i}in Delta }

${displaystyle X_{i}in Delta }$ . He comes

{displaystyle leftlangle domega ^{j},X_{i}wedge X_{l}rightrangle =sum limits _{k}leftlangle df_{k}^{j},X_{i}rightrangle leftlangle dg^{k},X_{l}rightrangle =0}

car

{displaystyle leftlangle dg^{k},X_{l}rightrangle =0}

${displaystyle leftlangle dg^{k},X_{l}rightrangle =0}$ . This proves (III).

Remarks [ modifier | Modifier and code ]

The equivalence (i) ⇔ (ii) of the geometric formulation of the Theorem of Frobenius extends to the infinite dimension by reasoning with bananachic varieties ^{[ 9 ]}. On the other hand, it does not extend in the case of the varieties of Fréchet.
In the case where ${displaystyle p=n-1}$
In the analytical case, Cartan-Kähler’s theorem (in) is a theorem of existence of an integral variety for a differential system; This theorem is a generalization of the Frobenius theorem.

Notes [ modifier | Modifier and code ]

References [ modifier | Modifier and code ]

(of) Wei-liang Chow , ‘ About systems of linear partial differential equations first order » , Mathematical annals , vol. 117, 1940-1941, p. 98-105 ( read online )
Jean Dieudonné , Analysis elements, vol. 1, 3 and 4 , Gauthier-Villars, 1969-1971
(in) Velimir Jurdjevic , Geometric Control Theory , Cambridge University Press, 1997
(in) Serge Lang , Fundamentals of Differential Geometry , Springer, 1999
Daniel One eyed , Differential calculation and geometry , PUF, 1982
(in) His Samelson , ‘ Differential forms, the Early Days; or the Stories of Deahna’s Theorem and of Volterra’s Theorem » , The American Mathematical Monthly , vol. 108, n ^O6, 2001 , p. 522-530 ( read online )
(in) Michael Spivak , (A Comprehensive Introduction to) Differential Geometry [Detail of editions] , flight. 1, chap. 6 and 7
(in) Hector J. Sussmann , ‘ Orbits of Families of Vector Fields and Integrability of Distributions » , Trans. Am. Math. Soc. , vol. 180, 1973 , p. 171-188 ( read online )

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Theorem of Frobenius (differential geometry) – Wikipedia

Frobenius theorem: “functional” formulation [ modifier | Modifier and code ]

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Example [ modifier | Modifier and code ]

Frobenius theorem: geometric formulation [ modifier | Modifier and code ]

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