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The problem of describing the integral manifolds of maximal dimension for a [[Pfaffian system|Pfaffian system]] of Pfaffian equations
 
The problem of describing the integral manifolds of maximal dimension for a [[Pfaffian system|Pfaffian system]] of Pfaffian equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p0725301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\theta  ^  \alpha  = 0 ,\ \
 +
\alpha = 1 \dots q ,
 +
$$
  
given by a collection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p0725302.png" /> differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p0725303.png" />-forms which are linearly independent at each point in a certain domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p0725304.png" /> (or on a certain manifold). A submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p0725305.png" /> is called an integral manifold of the system (*) if the restrictions of the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p0725306.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p0725307.png" /> are identically zero. The problem was posed by J. Pfaff (1814).
+
given by a collection of $  q $
 +
differential $  1 $-
 +
forms which are linearly independent at each point in a certain domain $  M \subset  \mathbf R  ^ {n} $(
 +
or on a certain manifold). A submanifold $  N \subset  M $
 +
is called an integral manifold of the system (*) if the restrictions of the forms $  \theta  ^  \alpha  $
 +
to $  N $
 +
are identically zero. The problem was posed by J. Pfaff (1814).
  
From a geometric point of view the system (*) determines an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p0725308.png" />-dimensional distribution (a [[Pfaffian structure|Pfaffian structure]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p0725309.png" />, that is, a field
+
From a geometric point of view the system (*) determines an $  ( n - q ) $-
 +
dimensional distribution (a [[Pfaffian structure|Pfaffian structure]]) on $  M $,  
 +
that is, a field
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253010.png" /></td> </tr></table>
+
$$
 +
x  \mapsto  P _ {x}  = \
 +
\{ {y \in \mathbf R  ^ {n} } : {\theta _ {x}  ^  \alpha  ( y) = 0 } \}
 +
,\ \
 +
x \in M ,
 +
$$
  
of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253011.png" />-dimensional subspaces, and the Pfaffian problem consists of describing the submanifolds of maximum possible dimension tangent to this field. The importance of the Pfaffian problem lies in the fact that the integration of an arbitrary partial differential equation can be reduced to a Pfaffian problem. For example, the integration of a first-order equation
+
of $  ( n - q ) $-
 +
dimensional subspaces, and the Pfaffian problem consists of describing the submanifolds of maximum possible dimension tangent to this field. The importance of the Pfaffian problem lies in the fact that the integration of an arbitrary partial differential equation can be reduced to a Pfaffian problem. For example, the integration of a first-order equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253012.png" /></td> </tr></table>
+
$$
 +
F \left ( x  ^ {i} , u ,
 +
\frac{\partial  u }{\partial  x  ^ {i} }
 +
\right )  = 0
 +
$$
  
reduces to the Pfaffian problem for the Pfaffian equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253013.png" /> on the submanifold (generally speaking with singularities) of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253014.png" /> defined by the equation
+
reduces to the Pfaffian problem for the Pfaffian equation $  \theta = d u - p _ {i}  d x  ^ {i} = 0 $
 +
on the submanifold (generally speaking with singularities) of the space $  \mathbf R  ^ {2n+} 1 $
 +
defined by the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253015.png" /></td> </tr></table>
+
$$
 +
F ( x  ^ {i} , u , p _ {i} )  = 0 .
 +
$$
  
A completely-integrable [[Pfaffian system|Pfaffian system]] (and also a single [[Pfaffian equation|Pfaffian equation]] of constant class) can be locally reduced to a simple canonical form. In these cases the solution of the Pfaffian problem reduces to the solution of ordinary differential equations. In the general case (in the class of smooth functions) the Pfaffian problem has not yet been solved (1989). The Pfaffian problem was solved by E. Cartan in the analytic case in his theory of systems in involution (cf. [[Involutional system|Involutional system]]). The formulation of the basic theorem of Cartan is based on the concept of a regular integral element. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253016.png" />-dimensional subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253017.png" /> of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253018.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253020.png" />-dimensional integral element of the system (*) if
+
A completely-integrable [[Pfaffian system|Pfaffian system]] (and also a single [[Pfaffian equation|Pfaffian equation]] of constant class) can be locally reduced to a simple canonical form. In these cases the solution of the Pfaffian problem reduces to the solution of ordinary differential equations. In the general case (in the class of smooth functions) the Pfaffian problem has not yet been solved (1989). The Pfaffian problem was solved by E. Cartan in the analytic case in his theory of systems in involution (cf. [[Involutional system|Involutional system]]). The formulation of the basic theorem of Cartan is based on the concept of a regular integral element. A $  k $-
 +
dimensional subspace $  E _ {k} $
 +
of the tangent space $  T _ {x} M $
 +
is called a $  k $-
 +
dimensional integral element of the system (*) if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253021.png" /></td> </tr></table>
+
$$
 +
\theta  ^  \alpha  ( E _ {k} )  = 0 ,\ \
 +
d \theta  ^  \alpha  ( E _ {k} \wedge E _ {k} )  = 0 ,\  \alpha = 1 \dots q .
 +
$$
  
The subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253022.png" /> of the cotangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253023.png" /> generated by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253024.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253028.png" /> is the operation of interior multiplication (contraction), is called the polar system of the integral element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253029.png" />. The integral element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253030.png" /> is regular if there exists a flag <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253031.png" /> for which
+
The subspace $  S ( E _ {k} ) $
 +
of the cotangent space $  T _ {r}  ^ {*} M $
 +
generated by the $  1 $-
 +
forms $  \theta  ^  \alpha  \mid  _ {x} $,  
 +
$  ( v  \llcorner d \theta  ^  \alpha  ) \mid  _ {x} $,  
 +
where $  v \in E _ {k} $
 +
and $  \llcorner $
 +
is the operation of interior multiplication (contraction), is called the polar system of the integral element $  E _ {k} $.  
 +
The integral element $  E _ {k} $
 +
is regular if there exists a flag $  E _ {k} \supset {} \dots \supset E _ {1} \supset 0 $
 +
for which
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253032.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm dim}  E _ {i}  = i ,\ \
 +
\mathop{\rm dim}  S ( E _ {i} )  = {\max  \mathop{\rm dim} }  S ( E _ {i}  ^  \prime  ) ,
 +
$$
  
where the maximum is taken over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253033.png" />-dimensional integral elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253034.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253035.png" />. Cartan's theorem asserts the following: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253036.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253037.png" />-dimensional integral manifold of a Pfaffian system with analytic coefficients and let, for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253038.png" />, the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253039.png" /> be a regular integral element. Then for any integral element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253040.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253041.png" /> there exists in a certain neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253042.png" /> an integral manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253043.png" />, locally containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253044.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253045.png" />. Cartan's theorem has been generalized to arbitrary differential systems given by ideals in the algebra of differential forms on a manifold (the Cartan–Kähler theorem).
+
where the maximum is taken over all $  i $-
 +
dimensional integral elements $  E _ {i}  ^  \prime  $
 +
containing $  E _ {i-} 1 $.  
 +
Cartan's theorem asserts the following: Let $  N $
 +
be a $  k $-
 +
dimensional integral manifold of a Pfaffian system with analytic coefficients and let, for a certain $  x \in N $,  
 +
the tangent space $  T _ {x} N $
 +
be a regular integral element. Then for any integral element $  E _ {k+} 1 \supset T _ {x} N $
 +
of dimension $  k + 1 $
 +
there exists in a certain neighbourhood of the point $  x $
 +
an integral manifold $  \widetilde{N}  $,  
 +
locally containing $  N $,  
 +
for which $  E _ {k+} 1 = T _ {x} \widetilde{N}  $.  
 +
Cartan's theorem has been generalized to arbitrary differential systems given by ideals in the algebra of differential forms on a manifold (the Cartan–Kähler theorem).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Cartan, "Sur la théorie des systèmes en involution et ses applications à la relativité" ''Bull. Soc. Math. France'' , '''59''' (1931) pp. 88–118 {{MR|1504975}} {{ZBL|0002.26401}} {{ZBL|57.0551.02}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E. Cartan, "Leçons sur les invariants intégraux" , Hermann (1922) {{MR|0355764}} {{ZBL|48.0538.02}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P.K. Rashevskii, "Geometric theory of partial differential equations" , Moscow-Leningrad (1947) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) {{MR|0193578}} {{ZBL|0129.13102}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> P.A. Griffiths, "Exterior differential systems and the calculus of variations" , Birkhäuser (1983) {{MR|0684663}} {{ZBL|0512.49003}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Cartan, "Sur la théorie des systèmes en involution et ses applications à la relativité" ''Bull. Soc. Math. France'' , '''59''' (1931) pp. 88–118 {{MR|1504975}} {{ZBL|0002.26401}} {{ZBL|57.0551.02}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E. Cartan, "Leçons sur les invariants intégraux" , Hermann (1922) {{MR|0355764}} {{ZBL|48.0538.02}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P.K. Rashevskii, "Geometric theory of partial differential equations" , Moscow-Leningrad (1947) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) {{MR|0193578}} {{ZBL|0129.13102}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> P.A. Griffiths, "Exterior differential systems and the calculus of variations" , Birkhäuser (1983) {{MR|0684663}} {{ZBL|0512.49003}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
==Pfaffian problems and partial differential equations.==
 
==Pfaffian problems and partial differential equations.==
 
Let
 
Let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253046.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
F _ {h} \left ( x  ^ {i} , u  ^ {j} ,
 +
\frac{\partial  ^  \alpha  }{\partial  x  ^  \alpha  }
 +
u  ^ {k} \right )  = 0 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253047.png" /></td> </tr></table>
+
$$
 +
= 1 \dots p,\  i  = 1 \dots n,\  j  = 1 \dots m,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253048.png" /></td> </tr></table>
+
$$
 +
\alpha  = ( a _ {1} \dots a _ {n} ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253049.png" /></td> </tr></table>
+
$$
 +
| \alpha |  = a _ {1} + \dots + a _ {n}  \leq  r,\  a _ {i}  \in  \{ 0, 1, . . . \} ,
 +
$$
  
be a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253050.png" /> partial differential equations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253051.png" /> functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253052.png" /> variables of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253053.png" />. Introduce the variables
+
be a system p $
 +
partial differential equations for $  m $
 +
functions in $  n $
 +
variables of order $  \leq  r $.  
 +
Introduce the variables
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253054.png" /></td> </tr></table>
+
$$
 +
p ^ {\alpha , k } ,\ \
 +
1 \leq  | \alpha | \leq  r,\ \
 +
k = 1 \dots m .
 +
$$
  
 
Replacing the equations (a1) with the equations
 
Replacing the equations (a1) with the equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a2 }
 +
\widetilde{F}  {} _ {h} ( x  ^ {i} , u  ^ {j} , p ^ {\alpha ,k } )  = 0
 +
$$
  
 
and adding to this the Pfaffian system
 
and adding to this the Pfaffian system
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253056.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
$$ \tag{a3 }
 +
dp ^ {\alpha , k } - \sum _ { i= } 1 ^ { n }
 +
p ^ {\alpha ( i), k }  d x  ^ {i}  = 0 ,\ \
 +
0 \leq  | \alpha | \leq  r- 1 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253058.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253059.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253060.png" />, one finds a system (a2)–(a3) of equations which are equivalent to equations (a1) in a suitable sense. Thus, if (locally) (a2) defines a subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253061.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253062.png" />-space, then a solution of the Pfaffian problem (a3) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253063.png" /> defines a solution of (a1) in the sense that the projection onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253064.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253065.png" /> as the case may be) gives the graph of a solution of (a1).
+
where $  p  ^ {0,k} = u  ^ {k} $
 +
and $  \alpha  ^ {(} i) = ( a _ {1} \dots a _ {i-} 1 , a _ {i} + 1 , a _ {i+} 1 \dots a _ {n} ) $
 +
if $  \alpha = ( a _ {1} \dots a _ {n} ) $
 +
for $  i = 1 \dots n $,  
 +
one finds a system (a2)–(a3) of equations which are equivalent to equations (a1) in a suitable sense. Thus, if (locally) (a2) defines a subvariety $  M $
 +
in $  ( x  ^ {i} , u  ^ {j} , p ^ {\alpha , k } ) $-
 +
space, then a solution of the Pfaffian problem (a3) on $  M $
 +
defines a solution of (a1) in the sense that the projection onto $  \mathbf R  ^ {n} \times \mathbf R  ^ {m} $(
 +
or $  \mathbf C  ^ {n} \times \mathbf C  ^ {m} $
 +
as the case may be) gives the graph of a solution of (a1).
  
 
For instance, in the case of a single second-order equation
 
For instance, in the case of a single second-order equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253066.png" /></td> </tr></table>
+
$$
 +
F \left ( x  ^ {1} , x  ^ {2} , u ,\
 +
 
 +
\frac{\partial  u }{\partial  x  ^ {1} }
 +
,
 +
\frac{\partial  u }{\partial  x  ^ {2} }
 +
,\
 +
 
 +
\frac{\partial  ^ {2} u }{\partial  x  ^ {1} \partial  x  ^ {1} }
 +
,\
 +
 
 +
\frac{\partial  ^ {2} u }{\partial  x  ^ {1} \partial  x  ^ {2} }
 +
,\
 +
 
 +
\frac{\partial  ^ {2} u }{\partial  x  ^ {2} \partial  x  ^ {2} }
 +
\right )  = 0
 +
$$
  
 
one has for (a2) and (a3), respectively,
 
one has for (a2) and (a3), respectively,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253067.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2prm)</td></tr></table>
+
$$ \tag{a2\prime }
 +
\widetilde{F}  ( x  ^ {1} , x  ^ {2} , u , p  ^ {1} , p  ^ {2} ,\
 +
p  ^ {11} , p  ^ {12} , p  ^ {22} )  = 0,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253068.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3prm)</td></tr></table>
+
$$ \tag{a3\prime }
 +
\left . \begin{array}{c}
 +
du  = p  ^ {1}  dx  ^ {1} + p  ^ {2}  dx  ^ {2} ,
 +
\\
 +
dp  ^ {1}  = p  ^ {11}  dx  ^ {1} + p  ^ {12}  dx  ^ {2} , \\
 +
dp
 +
^ {2}  = p  ^ {12}  dx  ^ {1} + p ^ {22}  dx  ^ {2}
 +
\end{array}
 +
\right \} .
 +
$$
  
The main equations are (a2); the remaining equations (a3) express that the solutions of (a2) of interest are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253069.png" />-jets (cf. [[Jet|Jet]] and [[Partial differential equations on a manifold|Partial differential equations on a manifold]]) of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253070.png" />. This leads to the idea of a system of partial differential equations on a manifold of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253071.png" /> as being determined by a set of functions on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253072.png" />-th jet bundle; cf. [[Partial differential equations on a manifold|Partial differential equations on a manifold]] for more details.
+
The main equations are (a2); the remaining equations (a3) express that the solutions of (a2) of interest are $  r $-
 +
jets (cf. [[Jet|Jet]] and [[Partial differential equations on a manifold|Partial differential equations on a manifold]]) of functions $  \mathbf R  ^ {n} \rightarrow \mathbf R  ^ {m} $.  
 +
This leads to the idea of a system of partial differential equations on a manifold of order $  r $
 +
as being determined by a set of functions on the $  r $-
 +
th jet bundle; cf. [[Partial differential equations on a manifold|Partial differential equations on a manifold]] for more details.
  
In the setting of equations like (a2), (a3) the following generalization of Frobenius' theorem on complete integrability is of interest. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253073.png" /> be a set of differential forms on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253075.png" /> a set of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253076.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253077.png" /> be such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253079.png" />. Suppose that
+
In the setting of equations like (a2), (a3) the following generalization of Frobenius' theorem on complete integrability is of interest. Let $  \omega  ^ {1} , \dots , \omega  ^ {r} $
 +
be a set of differential forms on a manifold $  M $
 +
and $  f ^ { 1 } , \dots , f ^ { s } $
 +
a set of functions on $  M $.  
 +
Let $  m \in M $
 +
be such that $  f ^ { i } ( m) = 0 $,  
 +
$  i = 1 \dots s $.  
 +
Suppose that
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253081.png" /> are in the ideal of differential forms generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253082.png" />;
+
i) $  d \omega  ^ {i} $
 +
and $  df ^ { j } $
 +
are in the ideal of differential forms generated by $  \omega  ^ {1} , \dots , \omega  ^ {r} ;  f ^ { 1 } \dots f ^ { s } $;
  
ii) the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253083.png" /> are linearly independent at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253084.png" />.
+
ii) the $  \omega  ^ {i} $
 +
are linearly independent at $  m $.
  
(Recall that the linearly independent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253085.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253086.png" /> form an involutive system if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253087.png" /> is in the ideal generated by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253088.png" />, cf. [[Involutive distribution|Involutive distribution]].) Then there is a unique germ of a submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253089.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253090.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253092.png" />, such that the differential forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253093.png" /> and functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253094.png" /> restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253095.png" /> are zero. Further if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253096.png" /> are functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253097.png" /> near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253098.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p07253099.png" /> are linearly independent at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530100.png" />, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530101.png" /> give a coordinate chart of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530102.png" /> near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530103.png" />.
+
(Recall that the linearly independent $  1 $-
 +
forms $  \omega  ^ {1} \dots \omega  ^ {r} $
 +
form an involutive system if $  d \omega  ^ {i} $
 +
is in the ideal generated by the $  \omega  ^ {i} $,  
 +
cf. [[Involutive distribution|Involutive distribution]].) Then there is a unique germ of a submanifold $  N $
 +
at $  m $
 +
of dimension $  n $,  
 +
$  r+ n = \mathop{\rm dim}  M $,  
 +
such that the differential forms $  \omega  ^ {i} $
 +
and functions $  f ^ { j } $
 +
restricted to $  N $
 +
are zero. Further if $  x  ^ {1} \dots x  ^ {n} $
 +
are functions on $  M $
 +
near $  m $
 +
such that $  \omega  ^ {1} \dots \omega  ^ {r} , dx  ^ {1} \dots dx  ^ {n} $
 +
are linearly independent at $  m $,  
 +
then the $  x  ^ {1} \dots x  ^ {n} $
 +
give a coordinate chart of $  N $
 +
near $  m $.
  
 
==Cartan–Kähler theorem for differential systems defined by ideals.==
 
==Cartan–Kähler theorem for differential systems defined by ideals.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530105.png" />, be a Pfaffian system on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530106.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530107.png" /> be an integral manifold of this system. Then obviously the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530109.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530110.png" /> is any differential form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530111.png" />, are also zero on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530112.png" />. Thus all the elements of the differential ideal generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530113.png" /> in the differential algebra of exterior differential forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530114.png" /> (cf. [[Differential form|Differential form]]; [[Differential ring|Differential ring]]) are zero on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530115.png" />. This leads to the idea of a differential system (of equations) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530116.png" /> as being defined by such an ideal. From now on let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530117.png" /> be a real [[Analytic manifold|analytic manifold]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530118.png" /> be the associated [[Sheaf|sheaf]] to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530119.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530120.png" /> is the sheaf of germs of rings of differential forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530121.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530122.png" /> be the sheaf of analytic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530123.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530124.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530125.png" />-module of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530126.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530127.png" />. A differential system on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530128.png" /> is a graded differential subsheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530129.png" /> of ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530130.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530131.png" /> (the ideal property), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530132.png" /> is generated by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530133.png" /> (the graded property) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530134.png" /> (the differential property). A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530135.png" />-dimensional integral manifold for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530136.png" /> is a submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530137.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530138.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530139.png" /> is zero. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530140.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530141.png" /> be the [[Grassmann manifold|Grassmann manifold]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530142.png" />-dimensional subspaces of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530143.png" />. The union of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530144.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530145.png" /> has a natural structure of a real-analytic manifold and the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530146.png" /> then defines a locally trivial fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530147.png" />. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530148.png" /> is called a contact element at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530149.png" />. Such an element is an integral element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530150.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530151.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530152.png" />; it is an integral element of a differential system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530153.png" /> if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530154.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530155.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530156.png" /> is an integral element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530157.png" />. An integral element of dimension zero (i.e. a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530158.png" />) is an [[Integral point|integral point]] (which is simply a solution of the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530159.png" /> for the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530160.png" />). The polar element of an integral element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530161.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530162.png" /> is the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530163.png" /> consisting of all vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530164.png" /> such that the span of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530165.png" /> is an integral element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530166.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530167.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530168.png" />, be the Grassmann coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530169.png" /> (cf. [[Exterior algebra|Exterior algebra]]; these are only defined up to a common scalar multiple). Now associate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530170.png" /> the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530171.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530172.png" />-modules in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530173.png" /> consisting of all the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530174.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530175.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530176.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530177.png" /> be the set of integral elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530178.png" /> (so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530179.png" /> is a certain subset of the Grassmann bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530180.png" />). The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530181.png" /> is called a regular integral element if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530182.png" /> is a regular local equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530183.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530184.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530185.png" /> is constant near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530186.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530187.png" />. Recall that a subsheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530188.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530189.png" /> is a manifold, is a regular local equation for (its set of zeros) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530190.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530191.png" /> if locally around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530192.png" /> there exist sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530193.png" /> such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530194.png" /> are linearly independent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530195.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530196.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530197.png" />.
+
Let $  \theta  ^ {a} = 0 $,  
 +
$  a = 1 \dots q $,  
 +
be a Pfaffian system on $  M $
 +
and let $  N $
 +
be an integral manifold of this system. Then obviously the $  d \theta  ^ {a} $
 +
and $  \theta  ^ {a} \wedge \omega $,  
 +
where $  \omega $
 +
is any differential form on $  M $,  
 +
are also zero on $  N $.  
 +
Thus all the elements of the differential ideal generated by $  \theta  ^ {1} \dots \theta  ^ {q} $
 +
in the differential algebra of exterior differential forms $  F( M) $(
 +
cf. [[Differential form|Differential form]]; [[Differential ring|Differential ring]]) are zero on $  N $.  
 +
This leads to the idea of a differential system (of equations) on $  M $
 +
as being defined by such an ideal. From now on let $  M $
 +
be a real [[Analytic manifold|analytic manifold]]. Let $  {\mathcal F} ( M) $
 +
be the associated [[Sheaf|sheaf]] to $  F( M) $,  
 +
i.e. $  {\mathcal F} ( M) $
 +
is the sheaf of germs of rings of differential forms on $  M $.  
 +
Let $  {\mathcal O} ( M) $
 +
be the sheaf of analytic functions on $  M $
 +
and let $  {\mathcal F} _ {p} ( M) $
 +
be the $  {\mathcal O} ( M) $-
 +
module of p $-
 +
forms on $  M $.  
 +
A differential system on $  M $
 +
is a graded differential subsheaf $  {\mathcal G} $
 +
of ideals of $  {\mathcal F} ( M) = {\mathcal F} $,  
 +
i.e. $  {\mathcal F} {\mathcal G} = {\mathcal G} = {\mathcal G} {\mathcal F} $(
 +
the ideal property), $  {\mathcal G} $
 +
is generated by the $  {\mathcal G} _ {p} = {\mathcal F} _ {p} \cap {\mathcal G} $(
 +
the graded property) and $  d {\mathcal G} \subset  {\mathcal G} $(
 +
the differential property). A p $-
 +
dimensional integral manifold for $  {\mathcal G} $
 +
is a submanifold $  N $
 +
of $  M $
 +
on which $  {\mathcal G} $
 +
is zero. For each $  m \in M $
 +
let $  \mathop{\rm Gr} _ {p} ( m) $
 +
be the [[Grassmann manifold|Grassmann manifold]] of p $-
 +
dimensional subspaces of the tangent space $  T _ {m} M $.  
 +
The union of the $  \mathop{\rm Gr} _ {p} ( m) $
 +
for $  m \in M $
 +
has a natural structure of a real-analytic manifold and the projection $  \mathop{\rm Gr} _ {p} ( m) \ni E _ {p} \rightarrow m $
 +
then defines a locally trivial fibre bundle $  \mathop{\rm Gr} _ {p} ( M) \rightarrow M $.  
 +
An element $  E _ {p} \in  \mathop{\rm Gr} _ {p} ( m) $
 +
is called a contact element at $  m $.  
 +
Such an element is an integral element of $  {\mathcal G} _ {p} $
 +
if $  \omega ( E _ {p} ) = 0 $
 +
for all $  \omega \in {\mathcal G} _ {p} $;  
 +
it is an integral element of a differential system $  {\mathcal G} $
 +
if for all $  E _ {q} \subset  E _ {p} $,  
 +
0 \leq  q \leq  p $,  
 +
$  E _ {q} $
 +
is an integral element of $  {\mathcal G} _ {p} $.  
 +
An integral element of dimension zero (i.e. a point of $  M $)  
 +
is an [[Integral point|integral point]] (which is simply a solution of the equations $  f( m) = 0 $
 +
for the functions $  f \in {\mathcal G} _ {0} $).  
 +
The polar element of an integral element $  E _ {p} $
 +
for $  {\mathcal G} $
 +
is the element $  P( E _ {p} ) \supset E _ {p} $
 +
consisting of all vectors $  v \in T _ {m} M $
 +
such that the span of $  v , E _ {p} $
 +
is an integral element of $  {\mathcal G} $.  
 +
Let $  z ^ {i _ {1} \dots i _ {p} } $,  
 +
$  1 \leq  i _ {1} < \dots < i _ {p} \leq  n $,  
 +
be the Grassmann coordinates of $  E _ {p} $(
 +
cf. [[Exterior algebra|Exterior algebra]]; these are only defined up to a common scalar multiple). Now associate to $  {\mathcal G} _ {p} $
 +
the sheaf $  {\mathcal G} _ {p}  ^ {0} $
 +
of $  {\mathcal O} ( M) $-
 +
modules in $  {\mathcal O} (  \mathop{\rm Gr} _ {p} ( M)) $
 +
consisting of all the functions $  \sum _ {i \leq  i _ {1}  < \dots < i _ {p} \leq  n } a _ {i _ {1}  \dots i _ {p} } z ^ {i _ {1} \dots i _ {p} } $
 +
for all p $-
 +
forms $  \sum _ {1 \leq  i _ {1}  < \dots < i _ {p} \leq  n } a _ {i _ {1}  \dots i _ {p} } dx ^ {i _ {1} } \wedge \dots \wedge dx ^ {i _ {p} } \in {\mathcal G} _ {p} $.  
 +
Let $  {\mathcal I} ( {\mathcal G} _ {p} ) $
 +
be the set of integral elements of $  {\mathcal G} _ {p} $(
 +
so that $  {\mathcal I} ( {\mathcal G} _ {p} ) $
 +
is a certain subset of the Grassmann bundle $  \mathop{\rm Gr} _ {p} ( M) $).  
 +
The element $  E _ {p} $
 +
is called a regular integral element if $  {\mathcal G} _ {p}  ^ {0} $
 +
is a regular local equation for $  {\mathcal I} ( {\mathcal G} _ {p} ) $
 +
at $  E _ {p} $
 +
and $  \mathop{\rm dim} ( P( E _ {p} )) $
 +
is constant near $  E _ {p} $
 +
on $  {\mathcal I} ( {\mathcal G} _ {p} ) $.  
 +
Recall that a subsheaf $  {\mathcal A} \subset  {\mathcal O} ( X) $,  
 +
where $  X $
 +
is a manifold, is a regular local equation for (its set of zeros) $  N \subset  X $
 +
at $  m \in N \subset  X $
 +
if locally around $  m $
 +
there exist sections $  s _ {1} \dots s _ {t} \in \Gamma ( U, {\mathcal O} ( X)) $
 +
such that the $  ds _ {1} \dots ds _ {t} $
 +
are linearly independent on $  U $
 +
and $  m  ^  \prime  \in N \cap U $
 +
if and only if $  s _ {1} ( m  ^  \prime  ) = \dots = s _ {t} ( m  ^  \prime  ) = 0 $.
  
The first Cartan–Kähler existence theorem is now as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530198.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530199.png" />-dimensional integral manifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530200.png" /> which defines a regular element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530201.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530202.png" />. Suppose that there is a submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530203.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530204.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530205.png" /> and of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530206.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530207.png" />. Then locally around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530208.png" /> there exists a unique integral manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530209.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530210.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530211.png" />.
+
The first Cartan–Kähler existence theorem is now as follows. Let $  N $
 +
be a p $-
 +
dimensional integral manifold of $  G $
 +
which defines a regular element $  T _ {m} N \subset  T _ {m} M $
 +
at $  m \in N \subset  M $.  
 +
Suppose that there is a submanifold $  M  ^  \prime  $
 +
of $  M $
 +
containing $  N $
 +
and of dimension $  n+ p+ 1 -  \mathop{\rm dim}  P( T _ {m} N) $
 +
such that $  \mathop{\rm dim} ( T _ {m} M  ^  \prime  \cap P( T _ {m} N)) = p+ 1 $.  
 +
Then locally around $  m $
 +
there exists a unique integral manifold $  N  ^  \prime  $
 +
of dimension p+ 1 $
 +
contained in $  M  ^  \prime  $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530212.png" />, the only possible choice (locally) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530213.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530214.png" /> itself, and there is a unique integral manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530215.png" /> extending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530216.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530217.png" /> there is "one arbitrary function worth" freedom in choosing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530218.png" /> and one re-encounters the phenomenon that the solution of a partial differential equation may depend on arbitrary functions (such as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530219.png" /> with as solutions any function of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530220.png" />). The second Cartan–Kähler existence theorem, which is obtained by repeated application of the first, details the dependence on initial conditions and arbitrary functions.
+
If $  \mathop{\rm dim}  P( T _ {m} N) = p+ 1 $,  
 +
the only possible choice (locally) for $  M  ^  \prime  $
 +
is $  M $
 +
itself, and there is a unique integral manifold of dimension p+ 1 $
 +
extending $  N $.  
 +
If $  \mathop{\rm dim}  P( T _ {m} N) = p+ 2 $
 +
there is "one arbitrary function worth" freedom in choosing $  M  ^  \prime  $
 +
and one re-encounters the phenomenon that the solution of a partial differential equation may depend on arbitrary functions (such as $  u _ {x} = u _ {t} $
 +
with as solutions any function of the form $  \phi ( x+ t) $).  
 +
The second Cartan–Kähler existence theorem, which is obtained by repeated application of the first, details the dependence on initial conditions and arbitrary functions.
  
An immediate corollary of the first Cartan–Kähler existence theorem is as follows. Suppose one is given an integral element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530221.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530222.png" /> of the differential system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530223.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530224.png" /> which contains a regular integral element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530225.png" />. Then there exists (locally) an integral manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530226.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530227.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072530/p072530228.png" />.
+
An immediate corollary of the first Cartan–Kähler existence theorem is as follows. Suppose one is given an integral element $  E _ {p+} 1 $
 +
of dimension p+ 1 $
 +
of the differential system $  {\mathcal G} $
 +
at $  m \in M $
 +
which contains a regular integral element $  E _ {p} $.  
 +
Then there exists (locally) an integral manifold $  N $
 +
of dimension p+ 1 $
 +
such that $  T _ {m} N = E _ {p+} 1 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) pp. Chapt. V, Appendix 3 (Translated from French) {{MR|0882548}} {{ZBL|0643.53002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Cartan, "Les systèmes différentielles extérieurs et leur applications géométriques" , Hermann (1945)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E. Cartan, "Sur l'intégration des systèmes d'équations aux différentielles totales" ''Ann. Sci. Ec. Norm. Sup.'' , '''18''' (1901) pp. 241–311 {{MR|}} {{ZBL|32.0351.04}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E. Kähler, "Einführung in die Theorie der Systeme von Differentialgleichungen" , Teubner (1934) {{MR|}} {{ZBL|0011.16103}} {{ZBL|60.0401.08}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Kuranishi, "Lectures on exterior differential systems" , Tata Inst. (1962)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> J. Dieudonné, "Eléments d'analyse" , '''4''' , Gauthier-Villars (1977) pp. Chapt. XVIII, Sect. 13 {{MR|0467780}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) pp. Chapt. V, Appendix 3 (Translated from French) {{MR|0882548}} {{ZBL|0643.53002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Cartan, "Les systèmes différentielles extérieurs et leur applications géométriques" , Hermann (1945)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E. Cartan, "Sur l'intégration des systèmes d'équations aux différentielles totales" ''Ann. Sci. Ec. Norm. Sup.'' , '''18''' (1901) pp. 241–311 {{MR|}} {{ZBL|32.0351.04}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E. Kähler, "Einführung in die Theorie der Systeme von Differentialgleichungen" , Teubner (1934) {{MR|}} {{ZBL|0011.16103}} {{ZBL|60.0401.08}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Kuranishi, "Lectures on exterior differential systems" , Tata Inst. (1962)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> J. Dieudonné, "Eléments d'analyse" , '''4''' , Gauthier-Villars (1977) pp. Chapt. XVIII, Sect. 13 {{MR|0467780}} {{ZBL|}} </TD></TR></table>

Latest revision as of 08:06, 6 June 2020


The problem of describing the integral manifolds of maximal dimension for a Pfaffian system of Pfaffian equations

$$ \tag{* } \theta ^ \alpha = 0 ,\ \ \alpha = 1 \dots q , $$

given by a collection of $ q $ differential $ 1 $- forms which are linearly independent at each point in a certain domain $ M \subset \mathbf R ^ {n} $( or on a certain manifold). A submanifold $ N \subset M $ is called an integral manifold of the system (*) if the restrictions of the forms $ \theta ^ \alpha $ to $ N $ are identically zero. The problem was posed by J. Pfaff (1814).

From a geometric point of view the system (*) determines an $ ( n - q ) $- dimensional distribution (a Pfaffian structure) on $ M $, that is, a field

$$ x \mapsto P _ {x} = \ \{ {y \in \mathbf R ^ {n} } : {\theta _ {x} ^ \alpha ( y) = 0 } \} ,\ \ x \in M , $$

of $ ( n - q ) $- dimensional subspaces, and the Pfaffian problem consists of describing the submanifolds of maximum possible dimension tangent to this field. The importance of the Pfaffian problem lies in the fact that the integration of an arbitrary partial differential equation can be reduced to a Pfaffian problem. For example, the integration of a first-order equation

$$ F \left ( x ^ {i} , u , \frac{\partial u }{\partial x ^ {i} } \right ) = 0 $$

reduces to the Pfaffian problem for the Pfaffian equation $ \theta = d u - p _ {i} d x ^ {i} = 0 $ on the submanifold (generally speaking with singularities) of the space $ \mathbf R ^ {2n+} 1 $ defined by the equation

$$ F ( x ^ {i} , u , p _ {i} ) = 0 . $$

A completely-integrable Pfaffian system (and also a single Pfaffian equation of constant class) can be locally reduced to a simple canonical form. In these cases the solution of the Pfaffian problem reduces to the solution of ordinary differential equations. In the general case (in the class of smooth functions) the Pfaffian problem has not yet been solved (1989). The Pfaffian problem was solved by E. Cartan in the analytic case in his theory of systems in involution (cf. Involutional system). The formulation of the basic theorem of Cartan is based on the concept of a regular integral element. A $ k $- dimensional subspace $ E _ {k} $ of the tangent space $ T _ {x} M $ is called a $ k $- dimensional integral element of the system (*) if

$$ \theta ^ \alpha ( E _ {k} ) = 0 ,\ \ d \theta ^ \alpha ( E _ {k} \wedge E _ {k} ) = 0 ,\ \alpha = 1 \dots q . $$

The subspace $ S ( E _ {k} ) $ of the cotangent space $ T _ {r} ^ {*} M $ generated by the $ 1 $- forms $ \theta ^ \alpha \mid _ {x} $, $ ( v \llcorner d \theta ^ \alpha ) \mid _ {x} $, where $ v \in E _ {k} $ and $ \llcorner $ is the operation of interior multiplication (contraction), is called the polar system of the integral element $ E _ {k} $. The integral element $ E _ {k} $ is regular if there exists a flag $ E _ {k} \supset {} \dots \supset E _ {1} \supset 0 $ for which

$$ \mathop{\rm dim} E _ {i} = i ,\ \ \mathop{\rm dim} S ( E _ {i} ) = {\max \mathop{\rm dim} } S ( E _ {i} ^ \prime ) , $$

where the maximum is taken over all $ i $- dimensional integral elements $ E _ {i} ^ \prime $ containing $ E _ {i-} 1 $. Cartan's theorem asserts the following: Let $ N $ be a $ k $- dimensional integral manifold of a Pfaffian system with analytic coefficients and let, for a certain $ x \in N $, the tangent space $ T _ {x} N $ be a regular integral element. Then for any integral element $ E _ {k+} 1 \supset T _ {x} N $ of dimension $ k + 1 $ there exists in a certain neighbourhood of the point $ x $ an integral manifold $ \widetilde{N} $, locally containing $ N $, for which $ E _ {k+} 1 = T _ {x} \widetilde{N} $. Cartan's theorem has been generalized to arbitrary differential systems given by ideals in the algebra of differential forms on a manifold (the Cartan–Kähler theorem).

References

[1] E. Cartan, "Sur la théorie des systèmes en involution et ses applications à la relativité" Bull. Soc. Math. France , 59 (1931) pp. 88–118 MR1504975 Zbl 0002.26401 Zbl 57.0551.02
[2] E. Cartan, "Leçons sur les invariants intégraux" , Hermann (1922) MR0355764 Zbl 48.0538.02
[3] P.K. Rashevskii, "Geometric theory of partial differential equations" , Moscow-Leningrad (1947) (In Russian)
[4] S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) MR0193578 Zbl 0129.13102
[5] P.A. Griffiths, "Exterior differential systems and the calculus of variations" , Birkhäuser (1983) MR0684663 Zbl 0512.49003

Comments

Pfaffian problems and partial differential equations.

Let

$$ \tag{a1 } F _ {h} \left ( x ^ {i} , u ^ {j} , \frac{\partial ^ \alpha }{\partial x ^ \alpha } u ^ {k} \right ) = 0 , $$

$$ h = 1 \dots p,\ i = 1 \dots n,\ j = 1 \dots m, $$

$$ \alpha = ( a _ {1} \dots a _ {n} ) , $$

$$ | \alpha | = a _ {1} + \dots + a _ {n} \leq r,\ a _ {i} \in \{ 0, 1, . . . \} , $$

be a system $ p $ partial differential equations for $ m $ functions in $ n $ variables of order $ \leq r $. Introduce the variables

$$ p ^ {\alpha , k } ,\ \ 1 \leq | \alpha | \leq r,\ \ k = 1 \dots m . $$

Replacing the equations (a1) with the equations

$$ \tag{a2 } \widetilde{F} {} _ {h} ( x ^ {i} , u ^ {j} , p ^ {\alpha ,k } ) = 0 $$

and adding to this the Pfaffian system

$$ \tag{a3 } dp ^ {\alpha , k } - \sum _ { i= } 1 ^ { n } p ^ {\alpha ( i), k } d x ^ {i} = 0 ,\ \ 0 \leq | \alpha | \leq r- 1 , $$

where $ p ^ {0,k} = u ^ {k} $ and $ \alpha ^ {(} i) = ( a _ {1} \dots a _ {i-} 1 , a _ {i} + 1 , a _ {i+} 1 \dots a _ {n} ) $ if $ \alpha = ( a _ {1} \dots a _ {n} ) $ for $ i = 1 \dots n $, one finds a system (a2)–(a3) of equations which are equivalent to equations (a1) in a suitable sense. Thus, if (locally) (a2) defines a subvariety $ M $ in $ ( x ^ {i} , u ^ {j} , p ^ {\alpha , k } ) $- space, then a solution of the Pfaffian problem (a3) on $ M $ defines a solution of (a1) in the sense that the projection onto $ \mathbf R ^ {n} \times \mathbf R ^ {m} $( or $ \mathbf C ^ {n} \times \mathbf C ^ {m} $ as the case may be) gives the graph of a solution of (a1).

For instance, in the case of a single second-order equation

$$ F \left ( x ^ {1} , x ^ {2} , u ,\ \frac{\partial u }{\partial x ^ {1} } , \frac{\partial u }{\partial x ^ {2} } ,\ \frac{\partial ^ {2} u }{\partial x ^ {1} \partial x ^ {1} } ,\ \frac{\partial ^ {2} u }{\partial x ^ {1} \partial x ^ {2} } ,\ \frac{\partial ^ {2} u }{\partial x ^ {2} \partial x ^ {2} } \right ) = 0 $$

one has for (a2) and (a3), respectively,

$$ \tag{a2\prime } \widetilde{F} ( x ^ {1} , x ^ {2} , u , p ^ {1} , p ^ {2} ,\ p ^ {11} , p ^ {12} , p ^ {22} ) = 0, $$

$$ \tag{a3\prime } \left . \begin{array}{c} du = p ^ {1} dx ^ {1} + p ^ {2} dx ^ {2} , \\ dp ^ {1} = p ^ {11} dx ^ {1} + p ^ {12} dx ^ {2} , \\ dp ^ {2} = p ^ {12} dx ^ {1} + p ^ {22} dx ^ {2} \end{array} \right \} . $$

The main equations are (a2); the remaining equations (a3) express that the solutions of (a2) of interest are $ r $- jets (cf. Jet and Partial differential equations on a manifold) of functions $ \mathbf R ^ {n} \rightarrow \mathbf R ^ {m} $. This leads to the idea of a system of partial differential equations on a manifold of order $ r $ as being determined by a set of functions on the $ r $- th jet bundle; cf. Partial differential equations on a manifold for more details.

In the setting of equations like (a2), (a3) the following generalization of Frobenius' theorem on complete integrability is of interest. Let $ \omega ^ {1} , \dots , \omega ^ {r} $ be a set of differential forms on a manifold $ M $ and $ f ^ { 1 } , \dots , f ^ { s } $ a set of functions on $ M $. Let $ m \in M $ be such that $ f ^ { i } ( m) = 0 $, $ i = 1 \dots s $. Suppose that

i) $ d \omega ^ {i} $ and $ df ^ { j } $ are in the ideal of differential forms generated by $ \omega ^ {1} , \dots , \omega ^ {r} ; f ^ { 1 } \dots f ^ { s } $;

ii) the $ \omega ^ {i} $ are linearly independent at $ m $.

(Recall that the linearly independent $ 1 $- forms $ \omega ^ {1} \dots \omega ^ {r} $ form an involutive system if $ d \omega ^ {i} $ is in the ideal generated by the $ \omega ^ {i} $, cf. Involutive distribution.) Then there is a unique germ of a submanifold $ N $ at $ m $ of dimension $ n $, $ r+ n = \mathop{\rm dim} M $, such that the differential forms $ \omega ^ {i} $ and functions $ f ^ { j } $ restricted to $ N $ are zero. Further if $ x ^ {1} \dots x ^ {n} $ are functions on $ M $ near $ m $ such that $ \omega ^ {1} \dots \omega ^ {r} , dx ^ {1} \dots dx ^ {n} $ are linearly independent at $ m $, then the $ x ^ {1} \dots x ^ {n} $ give a coordinate chart of $ N $ near $ m $.

Cartan–Kähler theorem for differential systems defined by ideals.

Let $ \theta ^ {a} = 0 $, $ a = 1 \dots q $, be a Pfaffian system on $ M $ and let $ N $ be an integral manifold of this system. Then obviously the $ d \theta ^ {a} $ and $ \theta ^ {a} \wedge \omega $, where $ \omega $ is any differential form on $ M $, are also zero on $ N $. Thus all the elements of the differential ideal generated by $ \theta ^ {1} \dots \theta ^ {q} $ in the differential algebra of exterior differential forms $ F( M) $( cf. Differential form; Differential ring) are zero on $ N $. This leads to the idea of a differential system (of equations) on $ M $ as being defined by such an ideal. From now on let $ M $ be a real analytic manifold. Let $ {\mathcal F} ( M) $ be the associated sheaf to $ F( M) $, i.e. $ {\mathcal F} ( M) $ is the sheaf of germs of rings of differential forms on $ M $. Let $ {\mathcal O} ( M) $ be the sheaf of analytic functions on $ M $ and let $ {\mathcal F} _ {p} ( M) $ be the $ {\mathcal O} ( M) $- module of $ p $- forms on $ M $. A differential system on $ M $ is a graded differential subsheaf $ {\mathcal G} $ of ideals of $ {\mathcal F} ( M) = {\mathcal F} $, i.e. $ {\mathcal F} {\mathcal G} = {\mathcal G} = {\mathcal G} {\mathcal F} $( the ideal property), $ {\mathcal G} $ is generated by the $ {\mathcal G} _ {p} = {\mathcal F} _ {p} \cap {\mathcal G} $( the graded property) and $ d {\mathcal G} \subset {\mathcal G} $( the differential property). A $ p $- dimensional integral manifold for $ {\mathcal G} $ is a submanifold $ N $ of $ M $ on which $ {\mathcal G} $ is zero. For each $ m \in M $ let $ \mathop{\rm Gr} _ {p} ( m) $ be the Grassmann manifold of $ p $- dimensional subspaces of the tangent space $ T _ {m} M $. The union of the $ \mathop{\rm Gr} _ {p} ( m) $ for $ m \in M $ has a natural structure of a real-analytic manifold and the projection $ \mathop{\rm Gr} _ {p} ( m) \ni E _ {p} \rightarrow m $ then defines a locally trivial fibre bundle $ \mathop{\rm Gr} _ {p} ( M) \rightarrow M $. An element $ E _ {p} \in \mathop{\rm Gr} _ {p} ( m) $ is called a contact element at $ m $. Such an element is an integral element of $ {\mathcal G} _ {p} $ if $ \omega ( E _ {p} ) = 0 $ for all $ \omega \in {\mathcal G} _ {p} $; it is an integral element of a differential system $ {\mathcal G} $ if for all $ E _ {q} \subset E _ {p} $, $ 0 \leq q \leq p $, $ E _ {q} $ is an integral element of $ {\mathcal G} _ {p} $. An integral element of dimension zero (i.e. a point of $ M $) is an integral point (which is simply a solution of the equations $ f( m) = 0 $ for the functions $ f \in {\mathcal G} _ {0} $). The polar element of an integral element $ E _ {p} $ for $ {\mathcal G} $ is the element $ P( E _ {p} ) \supset E _ {p} $ consisting of all vectors $ v \in T _ {m} M $ such that the span of $ v , E _ {p} $ is an integral element of $ {\mathcal G} $. Let $ z ^ {i _ {1} \dots i _ {p} } $, $ 1 \leq i _ {1} < \dots < i _ {p} \leq n $, be the Grassmann coordinates of $ E _ {p} $( cf. Exterior algebra; these are only defined up to a common scalar multiple). Now associate to $ {\mathcal G} _ {p} $ the sheaf $ {\mathcal G} _ {p} ^ {0} $ of $ {\mathcal O} ( M) $- modules in $ {\mathcal O} ( \mathop{\rm Gr} _ {p} ( M)) $ consisting of all the functions $ \sum _ {i \leq i _ {1} < \dots < i _ {p} \leq n } a _ {i _ {1} \dots i _ {p} } z ^ {i _ {1} \dots i _ {p} } $ for all $ p $- forms $ \sum _ {1 \leq i _ {1} < \dots < i _ {p} \leq n } a _ {i _ {1} \dots i _ {p} } dx ^ {i _ {1} } \wedge \dots \wedge dx ^ {i _ {p} } \in {\mathcal G} _ {p} $. Let $ {\mathcal I} ( {\mathcal G} _ {p} ) $ be the set of integral elements of $ {\mathcal G} _ {p} $( so that $ {\mathcal I} ( {\mathcal G} _ {p} ) $ is a certain subset of the Grassmann bundle $ \mathop{\rm Gr} _ {p} ( M) $). The element $ E _ {p} $ is called a regular integral element if $ {\mathcal G} _ {p} ^ {0} $ is a regular local equation for $ {\mathcal I} ( {\mathcal G} _ {p} ) $ at $ E _ {p} $ and $ \mathop{\rm dim} ( P( E _ {p} )) $ is constant near $ E _ {p} $ on $ {\mathcal I} ( {\mathcal G} _ {p} ) $. Recall that a subsheaf $ {\mathcal A} \subset {\mathcal O} ( X) $, where $ X $ is a manifold, is a regular local equation for (its set of zeros) $ N \subset X $ at $ m \in N \subset X $ if locally around $ m $ there exist sections $ s _ {1} \dots s _ {t} \in \Gamma ( U, {\mathcal O} ( X)) $ such that the $ ds _ {1} \dots ds _ {t} $ are linearly independent on $ U $ and $ m ^ \prime \in N \cap U $ if and only if $ s _ {1} ( m ^ \prime ) = \dots = s _ {t} ( m ^ \prime ) = 0 $.

The first Cartan–Kähler existence theorem is now as follows. Let $ N $ be a $ p $- dimensional integral manifold of $ G $ which defines a regular element $ T _ {m} N \subset T _ {m} M $ at $ m \in N \subset M $. Suppose that there is a submanifold $ M ^ \prime $ of $ M $ containing $ N $ and of dimension $ n+ p+ 1 - \mathop{\rm dim} P( T _ {m} N) $ such that $ \mathop{\rm dim} ( T _ {m} M ^ \prime \cap P( T _ {m} N)) = p+ 1 $. Then locally around $ m $ there exists a unique integral manifold $ N ^ \prime $ of dimension $ p+ 1 $ contained in $ M ^ \prime $.

If $ \mathop{\rm dim} P( T _ {m} N) = p+ 1 $, the only possible choice (locally) for $ M ^ \prime $ is $ M $ itself, and there is a unique integral manifold of dimension $ p+ 1 $ extending $ N $. If $ \mathop{\rm dim} P( T _ {m} N) = p+ 2 $ there is "one arbitrary function worth" freedom in choosing $ M ^ \prime $ and one re-encounters the phenomenon that the solution of a partial differential equation may depend on arbitrary functions (such as $ u _ {x} = u _ {t} $ with as solutions any function of the form $ \phi ( x+ t) $). The second Cartan–Kähler existence theorem, which is obtained by repeated application of the first, details the dependence on initial conditions and arbitrary functions.

An immediate corollary of the first Cartan–Kähler existence theorem is as follows. Suppose one is given an integral element $ E _ {p+} 1 $ of dimension $ p+ 1 $ of the differential system $ {\mathcal G} $ at $ m \in M $ which contains a regular integral element $ E _ {p} $. Then there exists (locally) an integral manifold $ N $ of dimension $ p+ 1 $ such that $ T _ {m} N = E _ {p+} 1 $.

References

[a1] P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) pp. Chapt. V, Appendix 3 (Translated from French) MR0882548 Zbl 0643.53002
[a2] E. Cartan, "Les systèmes différentielles extérieurs et leur applications géométriques" , Hermann (1945)
[a3] E. Cartan, "Sur l'intégration des systèmes d'équations aux différentielles totales" Ann. Sci. Ec. Norm. Sup. , 18 (1901) pp. 241–311 Zbl 32.0351.04
[a4] E. Kähler, "Einführung in die Theorie der Systeme von Differentialgleichungen" , Teubner (1934) Zbl 0011.16103 Zbl 60.0401.08
[a5] M. Kuranishi, "Lectures on exterior differential systems" , Tata Inst. (1962)
[a6] J. Dieudonné, "Eléments d'analyse" , 4 , Gauthier-Villars (1977) pp. Chapt. XVIII, Sect. 13 MR0467780
How to Cite This Entry:
Pfaffian problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pfaffian_problem&oldid=24529
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article