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 or all real-analytic.

A fibred manifold is a triple consisting of two manifolds , and a differentiable mapping such that is surjective for all . An example is a vector bundle over . This means that locally around each the situation looks like the canonical projection (, ). A cross section over an open set is a differentiable mapping such that . An -jet of cross sections at is an equivalence class of cross sections defined by the following equivalence relation. Two cross sections , , are -jet equivalent at if and if for some (hence for all) coordinate systems around and one has

where , , . Let be the set of all -jets. In local coordinates looks like , . It readily follows that is a manifold with local coordinates , [a2], [a5]. The differentiable bundle is called the -th jet bundle of the fibred manifold . For the case of a vector bundle see also Linear differential operator; for the case one finds , the jet bundle of mappings . There are natural fibre bundle mappings for , defined in local coordinates by forgetting about the with . It is convenient to set and , and then is defined in the same way (forget about all and the ).

Let be the sheaf of (germs of) differentiable functions on . It is a sheaf of rings. A subsheaf of ideals of is a system of partial differential equations of order on . A solution of the system is a section such that for all . The set of integral points of (i.e. the zeros of on ) is denoted by . The prolongation of is defined as the system of order on generated by the (strictly speaking, the ) and the , , where on an jet at is defined by

In local coordinates the formal derivative is given by

where the sum on the right is over and all with , and , (and ).

The system is said to be involutive at an integral point , [a1], if the following two conditions are satisfied: i) is a regular local equation for the zeros of at (i.e. there are local sections of on an open neighbourhood of such that the integral points of in are precisely the points for which and are linearly independent at ); and ii) there is a neighbourhood of such that is a fibred manifold over (with projection ). For a system generated by linearly independent Pfaffian forms (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 be a system defined on , and suppose that is involutive at . Then there is a neighbourhood of satisfying the following. If and is in , then there is a solution of defined on a neighbourhood of such that at .

The Cartan–Kuranishi prolongation theorem says the following. Suppose that there exists a sequence of integral points of () projecting onto each other () such that: a) is a regular local equation for at ; and b) there is a neighbourhood of in such that its projection under contains a neighbourhood of in and such that is a fibred manifold. Then is involutive at for 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