Difference between revisions of "Partial differential equations on a manifold"
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− | <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> |
Revision as of 12:12, 27 September 2012
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 |
Partial differential equations on a manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Partial_differential_equations_on_a_manifold&oldid=18120