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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p0716401.png" /> or all real-analytic.
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A fibred manifold is a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p0716402.png" /> consisting of two manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p0716403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p0716404.png" /> and a differentiable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p0716405.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p0716406.png" /> is surjective for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p0716407.png" />. An example is a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p0716408.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p0716409.png" />. This means that locally around each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164010.png" /> the situation looks like the canonical projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164013.png" />). A cross section over an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164014.png" /> is a differentiable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164016.png" />. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164018.png" />-jet of cross sections at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164019.png" /> is an equivalence class of cross sections defined by the following equivalence relation. Two cross sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164021.png" />, are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164022.png" />-jet equivalent at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164023.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164024.png" /> and if for some (hence for all) coordinate systems around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164026.png" /> one has
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<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/p071/p071640/p07164027.png" /></td> </tr></table>
+
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.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164030.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164031.png" /> be the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164032.png" />-jets. In local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164033.png" /> looks like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164035.png" />. It readily follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164036.png" /> is a manifold with local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164037.png" />, [[#References|[a2]]], [[#References|[a5]]]. The differentiable bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164038.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164039.png" />-th jet bundle of the fibred manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164040.png" />. For the case of a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164041.png" /> see also [[Linear differential operator|Linear differential operator]]; for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164042.png" /> one finds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164043.png" />, the jet bundle of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164044.png" />. There are natural fibre bundle mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164045.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164046.png" />, defined in local coordinates by forgetting about the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164047.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164048.png" />. It is convenient to set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164050.png" />, and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164051.png" /> is defined in the same way (forget about all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164052.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164053.png" />).
+
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
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164054.png" /> be the [[Sheaf|sheaf]] of (germs of) differentiable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164055.png" />. It is a sheaf of rings. A subsheaf of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164056.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164057.png" /> is a system of partial differential equations of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164059.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164060.png" />. A solution of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164061.png" /> is a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164062.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164063.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164064.png" />. The set of integral points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164066.png" /> (i.e. the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164067.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164068.png" />) is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164069.png" />. The prolongation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164071.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164072.png" /> is defined as the system of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164073.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164074.png" /> generated by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164075.png" /> (strictly speaking, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164076.png" />) and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164078.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164079.png" /> on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164080.png" /> jet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164081.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164082.png" /> is defined by
+
$$
 +
\left .  
 +
\frac{\partial  ^  \alpha  s _ {1} }{\partial  x  ^  \alpha  }
 +
\
 +
\right | _ {x= x _ {0}  }  = \left .  
 +
\frac{\partial  ^  \alpha  s _ {2} }{\partial x  ^  \alpha  }
 +
\
 +
\right | _ {x= x _ {0}  } ,0 \leq  | \alpha | \leq  r ,
 +
$$
  
<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/p071/p071640/p07164083.png" /></td> </tr></table>
+
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} $).
  
In local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164084.png" /> the formal derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164085.png" /> is given by
+
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
  
<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/p071/p071640/p07164086.png" /></td> </tr></table>
+
$$
 +
( \partial  ^ {k} f  )( j _ {x}  ^ {r+1} ( s))  =
 +
\frac \partial {\partial  x  ^ {k} }
 +
f( j _ {x}  ^ {r} ( s)).
 +
$$
  
where the sum on the right is over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164087.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164088.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164089.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164091.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164092.png" />).
+
In local coordinates  $  ( x  ^ {i} , u  ^ {j} , p ^ {\alpha ,k } ) $
 +
the formal derivative  $  \partial  ^ {k} f $
 +
is given by
  
The system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164093.png" /> is said to be involutive at an integral point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164095.png" />, [[#References|[a1]]], if the following two conditions are satisfied: i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164096.png" /> is a regular local equation for the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164098.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p07164099.png" /> (i.e. there are local sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640100.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640101.png" /> on an open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640102.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640103.png" /> such that the integral points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640104.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640105.png" /> are precisely the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640106.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640108.png" /> are linearly independent at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640109.png" />); and ii) there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640110.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640111.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640112.png" /> is a fibred manifold over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640113.png" /> (with projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640114.png" />). For a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640115.png" /> generated by linearly independent Pfaffian forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640116.png" /> (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.
+
$$
 +
\partial  ^ {k} f ( x , u , p) =
 +
\frac{\partial  f }{\partial  x  ^ {k} }
 +
+ \sum p ^ {\alpha ( i),j }
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640117.png" /> be a system defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640118.png" />, and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640119.png" /> is involutive at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640120.png" />. Then there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640121.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640122.png" /> satisfying the following. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640123.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640124.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640125.png" />, then there is a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640126.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640127.png" /> defined on a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640128.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640129.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640130.png" />.
+
\frac{\partial  f }{\partial  p ^ {\alpha ,j } }
 +
,
 +
$$
  
The Cartan–Kuranishi prolongation theorem says the following. Suppose that there exists a sequence of integral points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640131.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640132.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640133.png" />) projecting onto each other (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640134.png" />) such that: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640135.png" /> is a regular local equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640136.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640137.png" />; and b) there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640138.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640139.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640140.png" /> such that its projection under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640141.png" /> contains a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640142.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640143.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640144.png" /> is a fibred manifold. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640145.png" /> is involutive at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640146.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071640/p071640147.png" /> 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]]].
+
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====
 
====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</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)</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</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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.W. Hirsch,  "Differential topology" , Springer  (1976)  pp. Sect. 2.4</TD></TR></table>
+
<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>

Latest revision as of 21:26, 3 January 2021


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) $, [a2], [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; 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 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 ) $, [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) this is equivalent to the involutiveness defined in Involutive distribution, [a2], [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 [a4].

References

[a1] M. Kuranishi, "On E. Cartan's prolongation theorem of exterior differential systems" Amer. J. Math. , 79 (1957) pp. 1–47 MR0081957
[a2] M. Kuranishi, "Lectures on involutive systems of partial differential equations" , Publ. Soc. Mat. São Paulo (1967) Zbl 0163.12001
[a3] 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 MR0217822 Zbl 0277.58008
[a4] M. Matsuda, "Cartan–Kuranishi's prolongation of differential systems combined with that of Lagrange–Jacobi" Publ. Math. RIMS , 3 (1967) pp. 69–84 MR222438
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How to Cite This Entry:
Partial differential equations on a manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Partial_differential_equations_on_a_manifold&oldid=18120