# Partial differential equations on a manifold

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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 $C ^ \infty$ 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].

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=51213