View source for Partial differential equations on a manifold

<!--
p0716401.png
$#A+1 = 141 n = 0
$#C+1 = 141 : ~/encyclopedia/old_files/data/P071/P.0701640 Partial differential equations on a manifold
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

The basic idea is that a partial differential equation is given by a set of functions in a jet bundle, which is natural because after all a (partial) differential equation is a relation between a function, its dependent variables and its derivatives up to a certain order. In the sequel, all manifolds and mappings are either all  $  C  ^  \infty  $
or all real-analytic.

A fibred manifold is a triple  $  ( M, N, \pi ) $
consisting of two manifolds  $  M $, 
$  N $
and a differentiable mapping  $  \pi :  M \rightarrow N $
such that  $  d \pi ( m) :  T _ {m} M \rightarrow T _ {\pi ( m) }  N $
is surjective for all  $  m \in M $. 
An example is a vector bundle  $  ( E, N, \pi ) $
over  $  N $. 
This means that locally around each  $  m \in M $
the situation looks like the canonical projection  $  \mathbf R  ^ {n} \times \mathbf R  ^ {m} \rightarrow \mathbf R  ^ {n} $(
$   \mathop{\rm dim}  M= m+ n $, 
$   \mathop{\rm dim}  N= n $). 
A cross section over an open set  $  U \subset  N $
is a differentiable mapping  $  s:  U \rightarrow \pi  ^ {-1} ( U) \subset  M $
such that  $  \pi \circ s =  \mathop{\rm id} $. 
An  $  r $-jet of cross sections at  $  x \in N $
is an equivalence class of cross sections defined by the following equivalence relation. Two cross sections  $  s _ {i} :  U _ {i} \rightarrow M $, 
$  i= 1, 2 $, 
are  $  r $-jet equivalent at  $  x _ {0} \in U _ {1} \cap U _ {2} $
if  $  s _ {1} ( x _ {0} ) = s _ {2} ( x _ {0} ) $
and if for some (hence for all) coordinate systems around  $  s _ {i} ( x _ {0} ) $
and  $  x _ {0} $
one has

$$ 
\left . 
\frac{\partial   ^  \alpha  s _ {1} }{\partial  x  ^  \alpha  }
 \ 
\right | _ {x= x _ {0}  }  =  \left . 
\frac{\partial   ^  \alpha  s _ {2} }{\partial  x  ^  \alpha  }
 \ 
\right | _ {x= x _ {0}  } ,\  0 \leq  | \alpha | \leq  r ,
$$

where  $  \alpha = ( a _ {1} \dots a _ {n} ) $, 
$  a _ {i} \in \{ 0, 1,\dots \} $, 
$  | \alpha | = a _ {1} + \dots + a _ {n} $. 
Let  $  J  ^ {r} ( \pi ) $
be the set of all  $  r $-jets. In local coordinates  $  \pi $
looks like  $  \mathbf R  ^ {n} \times \mathbf R  ^ {m} \rightarrow \mathbf R  ^ {n} $, 
$  ( x  ^ {1} \dots x  ^ {n} , u  ^ {1} \dots u  ^ {m} ) \rightarrow ( x  ^ {1} \dots x  ^ {n} ) $. 
It readily follows that  $  J  ^ {r} ( \pi ) $
is a manifold with local coordinates  $  ( x  ^ {i} , u  ^ {j} , p ^ {\alpha ,k } :  i= 1 \dots n;  j, k= 1 \dots m;   1 \leq  | \alpha | \leq  r) $, 
[[#References|[a2]]], [[#References|[a5]]]. The differentiable bundle  $  J  ^ {r} ( \pi ) $
is called the  $  r $-th jet bundle of the fibred manifold  $  \pi :  M \rightarrow N $. 
For the case of a vector bundle  $  E \rightarrow N $
see also [[Linear differential operator|Linear differential operator]]; for the case  $  \pi :  N \times N  ^  \prime  \rightarrow N $
one finds  $  J  ^ {r} ( N, N  ^  \prime  ) $, 
the jet bundle of mappings  $  N \rightarrow N  ^  \prime  $. 
There are natural fibre bundle mappings  $  \pi _ {r,k }  :  J  ^ {r} ( \pi ) \rightarrow J  ^ {k} ( \pi ) $
for  $  r \geq  k \geq  0 $, 
defined in local coordinates by forgetting about the  $  p  ^  \alpha  $
with  $  | \alpha | > k $. 
It is convenient to set  $  p ^ {0,k } = u  ^ {k} $
and  $  J  ^ {-1} ( \pi ) = N $, 
and then  $  \pi _ {r,- 1 }  :  J  ^ {r} ( \pi ) \rightarrow N $
is defined in the same way (forget about all  $  p  ^  \alpha  $
and the  $  u  ^ {j} $).

Let  $  {\mathcal O} ( J  ^ {r} ( \pi )) $
be the [[Sheaf|sheaf]] of (germs of) differentiable functions on  $  J  ^ {r} ( \pi ) $. 
It is a sheaf of rings. A subsheaf of ideals  $  \mathfrak a $
of  $  {\mathcal O}( J  ^ {r} ( \pi ) ) $
is a system of partial differential equations of order  $  r $
on  $  N $. 
A solution of the system  $  \mathfrak a $
is a section  $  s :  N \rightarrow M $
such that  $  f \circ J  ^ {r} ( s)= 0 $
for all  $  f \in \mathfrak a $. 
The set of integral points of  $  \mathfrak a $ (i.e. the zeros of  $  \mathfrak a $
on  $  J  ^ {r} ( \pi ) $) 
is denoted by  $  J ( \mathfrak a ) $. 
The prolongation  $  p ( \mathfrak a ) $
of  $  \mathfrak a $
is defined as the system of order  $  r+ 1 $
on  $  N $
generated by the  $  f \in \mathfrak a $(
strictly speaking, the  $  f \circ \pi _ {r,r- 1 }  $) 
and the  $  \partial   ^ {k} f $, 
$  f \in \mathfrak a $, 
where  $  \partial   ^ {k} f $
on an  $  r+ 1 $
jet  $  j _ {x}  ^ {r+1} ( s) $
at  $  x \in N $
is defined by

$$ 
( \partial   ^ {k} f  )( j _ {x}  ^ {r+1} ( s))  =  
\frac \partial {\partial  x  ^ {k} }
 f( j _ {x}  ^ {r} ( s)).
$$

In local coordinates  $  ( x  ^ {i} , u  ^ {j} , p ^ {\alpha ,k } ) $
the formal derivative  $  \partial   ^ {k} f $
is given by

$$ 
\partial   ^ {k} f ( x , u , p)  =  
\frac{\partial  f }{\partial  x  ^ {k} }
 + \sum p ^ {\alpha ( i),j }

\frac{\partial  f }{\partial  p ^ {\alpha ,j } }
 ,
$$

where the sum on the right is over  $  j= 1 \dots m $
and all  $  \alpha = ( a _ {1} \dots a _ {n} ) $
with  $  | \alpha | \leq  r $, 
and  $  \alpha ( i) = ( a _ {1} \dots a _ {i-1} , a _ {i} + 1 , a _ {i+1} \dots a _ {n} ) $, 
$  a _ {i} \in \{ 0, 1, \dots \} $ (and  $  p ^ {0,j } = u  ^ {j} $).

The system  $  \mathfrak a $
is said to be involutive at an integral point  $  z \in J  ^ {r} ( \pi ) $, 
[[#References|[a1]]], if the following two conditions are satisfied: i)  $  \mathfrak a $
is a regular local equation for the zeros of  $  \mathfrak a $
at  $  z $
(i.e. there are local sections  $  s _ {1} \dots s _ {t} \in \Gamma ( U , \mathfrak a ) $
of  $  \mathfrak a $
on an open neighbourhood  $  U $
of  $  z $
such that the integral points of  $  \mathfrak a $
in  $  U $
are precisely the points  $  z  ^  \prime  $
for which  $  s _ {j} ( z  ^  \prime  )= 0 $
and  $  ds _ {1} \dots ds _ {t} $
are linearly independent at  $  z $); 
and ii) there is a neighbourhood  $  U $
of  $  z $
such that  $  \pi _ {r+ 1,r }   ^ {-1} ( U) \cap J( p( \mathfrak a )) $
is a fibred manifold over  $  U \cap J ( \mathfrak a ) $(
with projection  $  \pi _ {r+ 1,r }  $). 
For a system  $  \mathfrak a $
generated by linearly independent Pfaffian forms  $  \theta  ^ {1} \dots \theta  ^ {k} $(
i.e. a Pfaffian system, cf. [[Pfaffian problem|Pfaffian problem]]) this is equivalent to the involutiveness defined in [[Involutive distribution|Involutive distribution]], [[#References|[a2]]], [[#References|[a3]]]. As in that case of involutiveness one has to deal with solutions.

Let  $  \mathfrak a $
be a system defined on  $  J  ^ {r} ( \pi ) $, 
and suppose that  $  \mathfrak a $
is involutive at  $  z \in J ( \mathfrak a ) $. 
Then there is a neighbourhood  $  U $
of  $  z $
satisfying the following. If  $  \widetilde{z}  \in J ( p  ^ {t} ( \mathfrak a )) $
and  $  \pi _ {r+ t,r }  ( \widetilde{z}  ) $
is in  $  U $, 
then there is a solution  $  f $
of  $  \mathfrak a $
defined on a neighbourhood of  $  x= \pi _ {r+ t,- 1 }  ( \widetilde{z}  ) $
such that  $  J ^ {r+ t } ( f  ) = \widetilde{z}  $
at  $  x $.

The Cartan–Kuranishi prolongation theorem says the following. Suppose that there exists a sequence of integral points  $  z  ^ {t} $
of  $  p  ^ {t} ( \mathfrak a ) $ ($  t= 0, 1,\dots $) 
projecting onto each other ( $  \pi _ {r+ t,r+ t- 1 }  ( z  ^ {t} ) = z  ^ {t-1} $) 
such that: a)  $  p  ^ {t} ( \mathfrak a ) $
is a regular local equation for  $  J( p  ^ {t} ( \mathfrak a )) $
at  $  z  ^ {t} $; 
and b) there is a neighbourhood  $  U  ^ {t} $
of  $  z  ^ {t} $
in  $  J( p  ^ {t} ( \mathfrak a ) ) $
such that its projection under  $  \pi _ {r+ t,r+ t- 1 }  $
contains a neighbourhood of  $  z  ^ {t-1} $
in  $  J ( p  ^ {t-1} ( \mathfrak a ) ) $
and such that  $  \pi _ {r+ t,r+ t- 1 }  :  U  ^ {t} \rightarrow \pi _ {r+ t,r+ t- 1 }  ( U  ^ {t} ) $
is a fibred manifold. Then  $  p  ^ {t} ( \mathfrak a ) $
is involutive at  $  z  ^ {t} $
for  $  t $
large enough. This prolongation theorem has important applications in the Lie–Cartan theory of infinite-dimensional Lie groups. The theorem has been extended to cover more general cases [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Kuranishi,   "On E. Cartan's prolongation theorem of exterior differential systems"  ''Amer. J. Math.'' , '''79'''  (1957)  pp. 1–47  {{MR|0081957}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Kuranishi,   "Lectures on involutive systems of partial differential equations" , Publ. Soc. Mat. São Paulo  (1967)  {{MR|}} {{ZBL|0163.12001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I.M. Singer,   S. Sternberg,   "The infinite groups of Lie and Cartan I. The transitive groups"  ''J. d'Anal. Math.'' , '''15'''  (1965)  pp. 1–114  {{MR|0217822}} {{ZBL|0277.58008}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Matsuda,   "Cartan–Kuranishi's prolongation of differential systems combined with that of Lagrange–Jacobi"  ''Publ. Math. RIMS'' , '''3'''  (1967)  pp. 69–84  {{MR|222438}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.W. Hirsch,   "Differential topology" , Springer  (1976)  pp. Sect. 2.4  {{MR|0448362}} {{ZBL|0356.57001}} </TD></TR></table>