Exterior differential systems

406 Pages·1991·2.687 MB·English

by  Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.H., Griffiths P.A.

INTRODUCTION
This bookgives a treatment of exterior differential systems. It willinclude both
the general theory and various applications.
Anexterior differential systemisasystem ofequationsonamanifolddefined by
equating to zero a number of exterior differential forms. When all the forms are
linear, it is called a pfaffian system. Our object is to study its integral manifolds,
i.e.,submanifolds satisfying allthe equations of the system. A fundamental fact is
thatevery equationimpliesthe oneobtainedbyexterior differentiation,sothatthe
complete set ofequations associated toan exterior differentialsystem constitutes a
differential idealin the algebraof allsmooth forms. Thus the theory is coordinate-
free and computations typically have an algebraic character; however, even when
coordinates are used in intermediate steps, the use of exterior algebra helps to
efficiently guidethe computations,and asa consequence the treatment adapts well
to geometrical and physical problems.
A system of partial differential equations, with any number of independent and
dependent variables and involving partial derivatives of any order, can be writ-
ten as an exterior differential system. In this case we are interested in integral
manifolds on which certain coordinates remain independent. The corresponding
notion in exterior differential systems is the independence condition: certain pfaf-
fian forms remain linearly independent. Partial differential equations and exterior
differentialsystems withanindependence conditionareessentiallythesameobject.
The latter, however, possess some advantages among which are the facts that the
formsthemselves oftenhaveageometricalmeaning,andthatthesymmetriesofthe
exterior differential system are larger than those generated simply by changes of
independent and dependent variables. Another advantage is that the coordinate-
free treatment naturally leads to the intrinsic features of many systems of partial
differential equations.
ItwasPfaffwhopioneeredthestudyofexteriordifferentialsystemsbyhisformu-
lation of the Pfaff problem in Pfaff [1814-15]. The exterior derivative of a pfaffian
form, called the bilinear covariant, was introduced by Frobenius in 1877 and effi-
cientlyused byDarbouxinDarboux[1882]. InhisbookCartan[1922],E´lieCartan
introducedexteriordifferentialformsofhigherdegree andtheirexteriorderivatives.
In 1904–08he was led to the notion of a pfaffian system in involution through his
work in generalizing the Maurer–Cartan forms in Lie groups to infinite Lie pseu-
dogroups. The geometrical concepts introduced in this study apply to general
exterior differential systems, as recognized by Goursat. An authoritative account
was given in Ka¨hler [1934],culminatingin an existence theorem now knownas the
Cartan–Ka¨hler theorem.
1
2 Introduction
On the side of partial differential equations the basic existence theorem is of
course the Cauchy–Kowalewski theorem. More general existence theorems were
given by Riquier [1910]. The use of differential operators in studying differential
geometry has been traditional and has an extensive literature.
Among our fundamental concepts are prolongation and involutivity. Intuitively
theformeristheclassicalwayofadjoiningthepartialderivativesthemselvesasnew
variables,andtakingasnewequationsthoseobtainedbydifferentiatingtheoldset,
while the latter is the property that further prolongations will not give essentially
new integrability conditions. Their precise definitions are more subtle, and will be
given in Chapters VI and IV respectively. A linear pfaffian system in involutionis
a “well-behaved” system.
This concept entered in correspondence between Cartan and Einstein [1979] on
relativity. While E´lie Cartan proved that the Einstein field equations in general
relativity based on distant parallelism form an involutive system, Einstein was at
first suspicious ofthe notion. Later he understood itand expressed his satisfaction
and appreciation.
Cartan expressed the involutivity condition in terms of certain integers, known
asCartan’stest. Inmodernlanguagethis isahomologicalcondition. Infact,Serre
provedin1963thatinvolutivityisequivalenttothevanishingofcertaincohomology
groups(see Guillemin–Sternberg[1964]). Thismakesitpossibletousethepowerful
tool of commutative algebra. At the very beginning one notices the similarity
between polynomialsand differential operators. It turns out that this relationship
goes much deeper, and the theory involves a mixture of both commutative and
exterior algebra.
A fundamental problem is whether a given differential system will,after a finite
numberofprolongationsleadtoaninvolutivesystem. Cartanattempted toanswer
thisquestion, butitwasKuranishi[1957]whofinallyprovedthe Cartan–Kuranishi
prolongationtheorem. Themaintoolishomologytheory. Aslightlyweakerversion
of the theorem will be proved in this book.
Weshouldhoweveremphasize thatdifferentialsystems notininvolutionarejust
as important. In fact, they are probably richer in content. For example, non-
generic conditions on a manifold such as isometric embedding in low codimension
or the presence of additional geometric structures frequently are expressed by a
non-involutivesystem. The last half of Cartan [1946]and Partie II of his “Œuvres
Compl`etes” (Cartan [1953]) are full of “examples”, many of which are topics in
their own right. An objective of this book is to call attention to these beautiful
results, which have so far been largely ignored.
As the results are coordinate-free, the theory applies well to global problems
and to non-linear problems. A guiding problem in the theory is the equivalence
problem:
Given two sets of linear differential forms θi, θ∗j in the coordinates xk, x∗l
respectively, 1 ≤ i,j,k,l ≤ n, both linearly independent, and given a Lie group
G⊂GL(n,R). To find the conditions that there are functions
x∗i =x∗i(x1,...,xn),
such that θ∗j, after the substitution ofthese functions,differ fromθi by atransfor-
mation of G. This gives rise to an exterior differential system. Cartan’s idea was
Introduction 3
to introduce the parameters of G as new variables, setting
(cid:1) (cid:1)
ωi = giθj, ω∗i = g∗iθ∗j, (gi), (g∗i)∈G.
j j j j
j
Then in the product of the manifoldwith G, the condition becomes
ωi =ω∗i,
which is symmetrical in both sides. This means we should study the problem in a
principal G-bundle and the formulationbecomes global.
The equivalence problem gives local Riemannian geometry when G = O(n)
and the local invariants of an almost complex structure when n = 2m and G =
GL(m,C). Similarly,itgivesthe localinvariantsofCR-geometrywhen n=2m−1
and G is a suitable subgroup of GL(m,C).
We have stated the equivalence problem because of its importance; it will not
be explicitly treated in this book; see Gardner [1989]for a modern exposition.
The subject is so rich that a worker in the field is torn between the devil of the
general theory andthe angelofgeometricalapplications,which present allkinds of
interesting phenomena. We have attempted to strike a balance. We will develop
the general theory both from the standpoint of exterior differential systems and
from that of partial differential equations. We will also give a large number of
applications. A summary of contents follows:
Chapter I gives a review of exterior algebra, with emphasis on results which are
relevant to exterior differential systems. For those who like an intrinsic treatment
it includes an introduction to jet bundles.
Chapter II treats some simple exterior differential systems, particularly those
which can be put in a normalform by a change of coordinates. They include com-
pletely integrable systems (Frobenius theorem) and the pfaffian equation. Cauchy
characteristics forexterior differentialsystems comeup naturally. Somearithmetic
invariants are introduced for pfaffian systems. Even a pfaffian system of codimen-
sion 2, only partially treated in the last section, is a rich subject, with several
interesting applications.
ChapterIIIdiscusses thegenerationofintegralmanifoldsthroughthesolutionof
a succession of initial-valueproblems. Various basic concepts are introduced. The
Cartan–Ka¨hler theorem is given as a generalization of the Cauchy–Kowalewsky
theorem; the proof followsthat of K¨ahler. As an applicationwe givea proofof the
isometric imbedding theorem of Cartan–Janet.
Chapter IV introduces the important concepts of involution, linear differential
systems,tableauandtorsion. Forlinearpfaffiansystemstheconditionofinvolution,
as expressed by Cartan’s test, takes a simple form that is useful in computing
examples. We also introduce the concept of prolongation,which willbe more fully
developed in Chapter VI.
As one example we show that the high-dimensionalCauchy–Riemannequations
areininvolution. Wealsostudythe system ofq partialdifferentialequationsofthe
secondorderforonefunctioninnvariablesandfindconditionsfortheirinvolutivity.
A geometrical application is made to the problem of isometric surfaces preserving
the lines of curvature. It is an example of an over-determined system which, after
several prolongations, leads to a simple and elegant result. In this example the
effectiveness of exterior differential systems is manifest.
4 Introduction
Chapter V introduces the characteristic variety of a differential system. Intu-
itivelyits tangent spaces are hyperplanes ofintegralelements whose extension fails
to be unique. It plays as important a role as the characteristics in classical par-
tial differential equations. For a linear pfaffian system a definition can be given in
terms of the symbol and the two agree in the absence of Cauchy characteristics.
We discuss in detail the case of surfaces in E3 and their Darboux frames, as this
exampleillustratesmanyofthe basicnotionsofexterior differentialsystems. Some
properties of the characteristic variety are given. The deeper ones require the sys-
temtobe involutiveandthe use ofthe complexcharacteristic variety. Theirproofs
rely on results of commutative algebra and are postponed to Chapter VIII.
ChapterVItreatsprolongation,anotherwell-knownprocess inthecaseofpartial
differential equations. The issue is whether any system with an independence con-
dition (I,Ω) can be prolonged to an involutive system in a finite number of steps
(Cartan–Kuranishi theorem). With our definition of prolongation we prove that
the first prolongation of an involutive linear pfaffian system is involutive, a result
that does not seem to appear in the literature. We establish a somewhat weaker
version of the Cartan–Kuranishi theorem, thus giving in a sense a positive answer
to the above question. As usual the general theory is illustrated by a number of
examples.
Chapter VII is devoted to some examples and applications. We give a classi-
fication of systems of first-order partial differential equations of two functions in
twovariables. Other examples include: triplyorthogonalsystems, finiteness ofweb
rank, isometric imbedding and the characteristic variety.
In Chapter VIII we study the algebra of a linear pfaffian system and its pro-
longations. The crucial information is contained in the tableau. Its properties are
givenbytheSpencer cohomologygroupsortheKoszulhomologygroups,whichare
dual to each other. Involutive tableau is characterized by the vanishing of certain
SpencercohomologyorKoszulhomologygroups. Itisaremarkablecoincidencethat
a regular integral flag and a quasi-regular graded SV-module represent essentially
the same object. Homologicalalgebra provides the tools to complete the proofs of
the theorems stated inChapters VandVI,andinparticularthe Cartan–Kuranishi
theorem. As a consequence sheaf theory in commutative algebra and micro-local
analysis in partialdifferential equations become paralleldevelopments.
ChaptersIXandXgiveanintroductiontotheSpencertheoryofover-determined
systems of partial differential equations. While Cartan began his theory in the
study of infinite pseudogroups, Spencer had a similar objective, viz., the study of
the deformations of pseudogroup structures. His approach is more in the spirit of
Lie, with a full use of modern concepts. We see in our account more emphasis
on the general theory, although many examples are given. We hope that after the
exposition in this book we come to realize that exterior differential systems and
partial differential equations are one and the same subject. It is conceivable that
different attires are needed for different purposes.
This book grew through our efforts to work through and appreciate Partie II
of Cartan’s “Œuvres Compl`etes” (Cartan [1953]), which we found to be full of
interesting ideas and details. Hopefully our presentation willhelp the study of the
originalwork,whichwe cannotreplace. Infact,forreaders who havegonethrough
most of this book we propose the followingproblem as afinal examination: Give a
report on his famous five-variablepaper, “Les syst`emes de Pfaff `a cinq variables et
les ´equations aux d´eriv´ees partielles du second ordre” (Cartan [1910]).
Introduction 5
æ
6 I.Preliminaries
CHAPTER I
PRELIMINARIES
Inthischapterwewillsetupsomenotationsandconventionsinexterioralgebra,
give a description of the basic topic of the book, and introduce the language of
jets which allows easy passage between partial differential equations and exterior
differentialsystems. Inparticularweestablishsomebasicresultsinexterioralgebra
Theorems 1.3,1.5, and 1.7 which willbe used in Chapter II.
§1. Review of Exterior Algebra.
Let V be a real vector space of dimension n and V∗ its dual space. An element
x ∈ V is called a vector and an element ω ∈ V∗ a covector. V and V∗ have a
pairing
(cid:4)x,ω(cid:5), x∈V, ω ∈V∗,
which is a real number and is linear in each of the arguments, x,ω.
Over V there is the exterior algebra, which is a graded algebra:
Λ(V)=Λ0(V)⊕Λ1(V)⊕···⊕Λn(V),
with
Λ0(V)=R, Λ1(V)=V.
An element ξ ∈Λ(V) can be written in a unique way as
ξ =ξ +ξ +···+ξ ,
0 1 n
where ξ ∈Λp(V) is called the p-th component of ξ. An element
p
ξ =ξ ∈Λp(V)
p
is called homogeneous of degree p or a multivector of dimension p.
Multiplication in Λ(V) will be denoted by the wedge sign: ∧. It is associative,
distributive, but not commutative. Instead it satisfies the relation
ξ∧η =(−1)pqη∧ξ, ξ ∈Λp(V), η ∈Λq(V).
The multivector ξ is called decomposable, if it can be written as a monomial
(1) ξ =x ∧···∧x , x ∈V.
1 p i
We have the followingfundamental fact:
§1. ReviewofExteriorAlgebra 7
Proposition1.1 (Criteriononlineardependence). The vectors x ,...,x are lin-
1 p
early dependent if and only if x ∧···∧x =0.
1 p
If the decomposable multivector ξ in(1) is not zero, then x ,...,x are linearly
1 p
independent and span alinear subspace W ofdimensionp inV. This space can be
described by
(2) W ={x∈V |x∧ξ =0}.
Let x(cid:2),...,x(cid:2) be another base in W. Then
1 p
ξ(cid:2) =x(cid:2) ∧···∧x(cid:2)
1 p
is a (non-zero) multiple of ξ. We will call ξ, defined up to a constant factor, the
Grassmann coordinate vector of W, and write
(3) [ξ]=W,
the bracket indicating the class of coordinate vectors differing from each other by
a non-zero factor.
In the same way there is over V∗ the exterior algebra
Λ(V∗)=Λ0(V∗)⊕Λ1(V∗)⊕···⊕Λn(V∗),
Λ0(V∗)=R, Λ1(V∗)=V∗.
An element of Λp(V∗) is called a form of degree p or simply a p-form.
Let e be a base of V and ωk its dual base, so that
i
(cid:4)e ,ωk(cid:5)=δk, 1≤i,k≤n.
i i
Then an element ξ ∈Λp(V) can be written
(cid:1)
(4) ξ =1/p! ai1...ipei1 ∧···∧eip
and an element α∈Λp(V∗) as
(cid:1)
(5) α=1/p! bi1...ipωi1 ∧···∧ωip.
In (4) and (5) the coefficients ai1...ip and bi1...ip are supposed to be anti-symmetric
in any two of their indices, so that they are well defined. It follows from (4) that
any multivector is a linear combinationof decomposable multivectors.
Forourapplicationsitisimportantto establishthe explicitdualityor pairingof
Λ(V) andΛ(V∗). Werequire thatΛp(V)andΛq(V∗),p(cid:8)=q,annihilateeachother.
Ittherefore suffices todefine the pairingofΛp(V)and Λp(V∗). Since, bythe above
remark, any multivector is a linear combination of decomposable multivectors, it
suffices to have the pairing of
ξ =x ∧···∧x , x ∈V,
1 p i
8 I.Preliminaries
and
α=ω1∧···∧ωp, ωi ∈V∗.
We will define
(6) (cid:4)ξ,α(cid:5)=det((cid:4)x ,ωk(cid:5)), 1≤i,k≤p.
i
Itcanbeimmediatelyverifiedthatthisdefinitionismeaningful,i.e.,ifξ (resp.α)is
expressed inadifferentwayasaproductofvectors(resp.covectors),theright-hand
side of (6) remains unchanged.
In terms of the expressions (4) and (5) the pairingis given by
(cid:1)
(7) (cid:4)ξ,α(cid:5)=1/p! ai1...ipbi1...ip.
Thisisprovedbyobservingthattheright-handsideof(7)islinearinthearguments
ξ and α and that the right-hand sides of both (6) and (7) are equal when ξ and α
are products of the elements of the dual bases.
An endomorphism f of the additive structure of Λ(V) is called a derivation of
degree k if it satisfies the conditions:
(i) f :ΛpV →Λp+kV, 0≤p≤n
(ii) f(ξ∧η)=f(ξ)∧η+(−1)kpξ∧f(η)
for ξ ∈ΛpV, η ∈ΛV.
A derivation of degree −1 is also called an anti-derivation.
Given ξ ∈V, we define the exterior product
e(ξ):Λ(V)→Λ(V)
by
e(ξ)η =ξ∧η η ∈Λ(V).
The adjoint operator of e(ξ),
ξ :Λ(V∗)→Λ(V∗)
is called the interior product, and is defined by the relation
(cid:4)η,ξ α(cid:5)=(cid:4)e(ξ)η,α(cid:5) η∈Λ(V),α∈Λ(V∗).
The followingresult is easily proved:
Proposition 1.2. If x∈V, then x is an anti-derivation.
Notice that e(x) is neither a derivation nor an anti-derivation.
Definition. A subring I ⊂Λ(V∗) is called an ideal, if:
a) α∈I implies α∧β ∈I for all β ∈Λ(V∗);
b) α∈I implies that all its components in Λ(V∗) are contained in I.
A subring satisfying the second condition is called homogeneous. As a conse-
quence of a) and b) we conclude that α ∈ I implies β ∧α ∈ I for all β ∈ Λ(V∗).
Thus all our ideals are homogeneous and two-sided.
GivenanidealI ⊂Λ(V∗),wewishtodetermine thesmallestsubspace W∗ ⊂V∗
suchthatI isgenerated,asanideal,byasetS ofelementsofΛ(W∗). Anelementof
§1. ReviewofExteriorAlgebra 9
I isthenasumofelementsoftheformσ∧β,σ ∈S,β ∈Λ(V∗). Ifx∈W =(W∗)⊥,
we have, since the interior product x is an anti-derivation,
x σ =0,
x (σ∧β)=±σ∧(x β)∈I.
Therefore we define
A(I)={x∈V |x I ⊂I},
where the last condition means that x α ∈ I, for all α ∈ I. A(I) is clearly a
subspace ofV. Itwillplaylateranimportantroleindifferentialsystems, forwhich
reason we willcall it the Cauchy characteristic space of I. Its annihilator
C(I)=A(I)⊥ ⊂V∗
willbe called the retracting subspace of I.
Theorem 1.3 (Retraction theorem). Let I be an ideal of Λ(V∗). Its retracting
subspace C(I) is the smallest subspace of V∗ such that Λ(C(I)) contains a set S of
elements generating I as an ideal. The set S also generates an ideal J in Λ(C(I)),
to be called a retracting ideal of I. There exists a mapping
∆:Λ(V∗)→Λ(C(I))
of graded algebras such that ∆(I)=J.
Proof. Suppose W∗ ⊂ V∗ be a subspace such that Λ(W∗) contains a set S of
elementswhichgenerate I asanideal. Bytheabovediscussion,ifx∈W =(W∗)⊥,
we have x I ⊂I. It followsthat W ⊂A(I), and consequently, C(I)=(A(I))⊥ ⊂
W∗.
Wenowchooseacomplementaryspace B ofC(I) inV∗,sothatV∗ =B⊕C(I).
Let ωi, 1≤i≤n, be a base in V∗ with
ω1,...,ωk ∈B, ωk+1,...,ωn∈C(I).
Its dual base e , 1 ≤ i ≤ n, then has the property that A(I) = {e ,...,e }. We
i 1 k
define
h :Λ(V∗)→Λ(V∗), 1≤j ≤k,
j
by
h (α)=α−ωj ∧(e α), α∈Λ(V∗).
j j
It is easy to verify that
h (α∧β)=h (α)∧h (β),
j j j
so that each h is a mapping of graded algebras. The same is therefore true of the
j
composition
∆=h ◦···◦h .
k 1
Since e ∈ A(I), 1 ≤ j ≤ k, we have h (I) ⊂ I, from which we get ∆(I) ⊂ I.
j j
Clearly we have the restrictions ∆| = 0, ∆| = Id. Since ∆ is a mapping of
B C(I)
graded algebras, this implies that Λ(C(I)) is the image of ∆.
10 I.Preliminaries
It remains to construct the set S in Λ(C(I)) which generates I. This is done by
induction on the degrees of the elements of I. Let I be the set of elements of I
p
of degree p. To exclude the trivial case that I = Λ(V∗) itself, we suppose I = ∅.
0
Using this assumption we have, by the definition of A(I), x α = (cid:4)x,α(cid:5) = 0,
x∈A(I), α∈I . It followsthat A(I) ⊂I⊥ of I ⊂C(I).
1 1 1
ToapplyinductionsupposethatI1,...,Ip−1aregeneratedbyelementsofΛ(C(I)).
Denote by Jp−1 the ideal generated by them. Recall that hj, 1 ≤j ≤ k, are map-
pings ofgraded algebras and induce the identity mappingon C(I). They therefore
leave Jp−1 invariant. By the definition of h1 we have h1(α)−α∈Jp−1. Applying
h2,...,hk successively, we get ∆(α)−α ∈ Jp−1. By replacing α by ∆(α) as a
generator of I, we complete the induction. (cid:1)
We wish to make some applications of the retraction theorem (Theorem 1.3).
First we recall that, dualizing (2) and (3), the Grassmann coordinate vector [α] of
a subspace W∗ ⊂ V∗ of dimension p is a non-zero decomposable p-covector such
that
W∗ ={ω∈V∗ |ω∧α=0}.
This notion can be extended to any p-form α, decomposable or not, by defining
L ={ω∈V∗ |ω∧α=0}.
α
L willbe called the space of linear divisors of α,because ofthe property given in
α
the followingtheorem:
Proposition1.4. Given a p-form α, let ω1,...,ωq be a base for L . Then α may
α
be written in the form
α=ω1∧···∧ωq ∧π, with π∈Λp−q(V∗).
Proof. Take first the case q = 1. We can suppose ω1 to be a base element of V∗.
By the expression (5) we can write
α=ω1∧π+α ,
1
where α does not involve ω1. The hypothesis ω1∧α= 0 implies α = 0, so that
1 1
the statement is true.
The general case followsby induction on q. (cid:1)
Theorem 1.5. Let I be an ideal generated by the linearly independent elements
ω1,...,ωs ∈ V∗ and the 2-form Ω ∈ Λ2(V∗). Let p be the smallest integer such
that
(8) Ωp+1∧ω1∧···∧ωs =0.
Then the retracting space C(I) is of dimension 2p + s and has the Grassmann
coordinate vector
Ωp∧ω1∧···∧ωs.
Proof. Consider first the case s=0. An element of I is a linear combination of Ω,
Ω2,...,Ωp (cid:8)=0. Hence byTheorem1.3,wehaveΩ∈Λ(C(I)),andΩp ∈Λ2p(C(I)).
The latter implies
dimC(I)≥2p.