Partial differential equations on a manifold

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
[a5]	M.W. Hirsch, "Differential topology" , Springer (1976) pp. Sect. 2.4 MR0448362 Zbl 0356.57001