Introduction to Exterior Differential Systems

204 Pages·2022·49.972 MB·English

by  Benjamin McKay

Benjamin McKay
Introduction
to Exterior
Differential
Systems
February 13, 2022
ThisworkislicensedunderaCreativeCommonsAttribution-ShareAlike4.0UnportedLicense.
iii

Preface
To the reader who wants to dip a toe in the water: read chapter 1. The
reader who continues on to chapters 2 and 4 will pick up the rest of the tools.
Subsequentchaptersprovethetheorems. Weassumethatthereaderisfamiliar
with elementary differential geometry on manifolds and with differential forms.
These lectures explain how to apply the Cartan–Kähler theorem to problems
in differential geometry. Given some differential equations, we want to decide
if they are locally solvable. The Cartan–Kähler theorem gives a linear algebra
test: if the equations pass the test, they are locally solvable. We give the
necessary background on partial differential equations in appendices A, B, and
the (not so necessary) background on moving frames in appendices D, F, G.
The reader should be aware of [4], which we will follow closely, and also the
canonical reference work [3] and the canonical textbook [19].
I wrote these notes for lectures at the Banach Center in Warsaw, at the
University of Paris Sud, and at the University of Rome Tor Vergata. I thank
Jan Gutt, Gianni Manno, Giovanni Moreno, Nefton Pali, and Filippo Bracci
for invitations to give those lectures, and Francesco Russo for the opportunity
to write these notes at the University of Catania.
v
Contents
1 Exterior differential systems 1
2 Tableaux 13
3 Example: almost complex structures 25
4 Prolongation 33
5 Cartan’s test 39
6 The characteristic variety 47
7 Proof of the Cartan–Kähler theorem 53
8 Cauchy characteristics 59
9 Example: conformal maps of surfaces 69
10 Example: Weingarten surfaces 73
A The Cauchy–Kovalevskaya theorem 77
B Symbol and characteristic variety 87
C Analytic convergence 97
D The moving frame 101
E Curvature of surfaces 115
F The Gauss–Bonnet theorem 125
G Geodesics on surfaces 133
Hints 149
Bibliography 191
Index 195
vi
Chapter 1
Exterior differential systems
We define exterior differential systems, and state the Cartan–Kähler theorem.
Background material from differential geometry
By the expression analytic, we mean real analytic. We assume that all of our
manifolds, maps, vector fields and differential forms are analytic. (In a few
exercises, we describe some such thing as smooth, i.e. C∞, not necessarily
analytic.) A submanifold is an immersion of manifolds, defined up to diffeomor-
phism of the domain. Denote the dimension of a manifold M by dimM. The
codimension of a submanifold S of M is dimM −dimS. A hypersurface in a
manifold M is a submanifold of codimension one. A hyperplane in a vector
space V is a linear subspace W ⊂V so that dim(V/W)=1.
Differential equations encoded in differential forms
To express a differential equation 0=f(cid:0)x,u,∂u(cid:1), add a variable p to represent
∂x
the derivative ∂u. Let ϑ ..= du − pdx, ω ..= dx. Let M be the manifold
∂x
M ..={(x,u,p)|0=f(x,u,p)} (assuming it is a manifold). Any submanifold
of M on which 0=ϑ and 06=ω is locally the graph of a solution. It is easy to
generalize this to any number of variables and any number of equations of any
order.
Exterior differential systems
An integral manifold of a collection of differential forms is a submanifold on
which the forms vanish. An exterior differential system on a manifold M is an
ideal I ⊂ Ω∗ of differential forms, closed under exterior derivative, splitting
into a direct sum
I =I1⊕···⊕IdimM
of forms of various degrees: Ik ..=I∩Ωk. Any collection of differential forms
has the same integral manifolds as the exterior differential system it generates.
1
2 Exterior differential systems
Some trivial examples: the exterior differential system generated by
a. 0,
b. Ω∗,
c. the pullbacks of all forms via a submersion,
d. dx1∧dy1+dx2∧dy2 in R4,
e. dy−zdx on R3.
1.1 What are the integral manifolds of our trivial examples?
By definition I0 = 0, i.e. there are no nonzero functions in I. If we wish
some functions to vanish, we can replace our manifold M by the zero locus
of those functions (which might not be a manifold, a technical detail we will
ignore).
Integral elements
An integral element of an exterior differential system I is a linear subspace of
a tangent space, on which all forms in I vanish. Every tangent space of any
integral manifold is an integral element, but some integral elements of some
exterior differential systems don’t arise as tangent spaces of integral manifolds.
A 1-dimensional integral element is an integral line. A 2-dimensional integral
element is an integral plane.
1.2 What are the integral elements of our trivial examples?
1.3 Supposethatthe1-formsinanexteriordifferentialsystemspanasubspace
of constant rank in each cotangent space. Prove that there is an integral curve
tangent to each integral line.
1.4 Give an example of an embedded integral manifold, whose every tangent
space is a hyperplane in a unique integral element, but which is not a hyper-
surface in an integral manifold.
1.5 Provethatak-dimensionallinearsubspaceofatangentspaceisanintegral
element of an exterior differential system I just when all k-forms in I vanish
on it.
The polar equations of an integral element E are the linear functions
w ∈T M 7→ϑ(w,e ,e ,...,e )
m 1 2 k
where ϑ ∈ Ik+1 and e ,e ,...,e ∈ E, for any k. They vanish on a vector
1 2 k
w just when the span of {w}∪E is an integral element. The set of polar
equations of E is a linear subspace of T∗M. If an integral element E lies in
m
Integral elements as points of the Grassmann bundle 3
a larger one F, then all polar equations of E occur among those of F: larger
integral elements have more (or at least the same) polar equations.
1.6 What are the polar equations of the integral elements of our trivial exam-
ples?
An integral flag of an exterior differential system is a sequence of nested
integral elements
0=E ⊂E ⊂E ⊂···⊂E
0 1 2 p
with dimensions 0,1,2,...,p. Danger: most authors require that a flag have
subspaces of all dimensions; we don’t: we only require that the subspaces have
all dimensions 0,1,2,...,p up to some dimension p. Successive subspaces have
successivelylargerspace ofpolarequations. The characters s ,s ,...,s ofthe
0 1 p
flag are the increments in rank of polar equations: E has polar equations of
k
rank s +s +···+s .
0 1 k
ConsiderallflagsinsideagivenE . Polarequationsremainlinearlyindepen-
p
dent under small motions of a flag. So if a flag in E has maximal dimensional
p
polar equations, i.e. character sums s ,s +s ,...,s +···+s , then so do all
0 0 1 0 p
nearby flags in E . The characters of the integral element E are those of any
p p
such flag.
1.7 What are the characters of the integral flags of our trivial examples?
Integral elements as points of the Grassmann bundle
The rank p Grassmann bundle of a manifold M is the set of all p-dimensional
linear subspaces of tangent spaces of M.
1.8 Recall how charts are defined on the Grassmann bundle. Prove that the
Grassmann bundle is a fiber bundle.
The integral elements of an exterior differential system I form a subset of
the Grassmann bundle. We would like to see if that subset is a submanifold,
and predict its dimension.
The Cartan–Kähler theorem
An integral element, say of dimension p and with characters s ,...,s , pre-
0 p
dicts the dimension dimM +s +2s +···+ps . It predicts correctly if the
1 2 p
nearby integral elements form a submanifold of the Grassmann bundle of the
predicted dimension. (We will see that they always sit in a submanifold of the
predicteddimension.)Anintegralelementwhichcorrectlypredictsdimensionis
involutive. An exterior differential system on a manifold M is involutive if each
component of M contains an involutive maximal integral element. (We will
see that involutivity of an exterior differential system implies that involutive
integral elements occur at a dense open subset of points of M.)
4 Exterior differential systems
Theorem 1.1 (Cartan–Kähler). Every involutive integral element of any ana-
lytic exterior differential system is tangent to an analytic integral manifold.
1.9 Giveanexampleofanexteriordifferentialsystemwhosecharactersarenot
constant.
Example: Lagrangian submanifolds
We employ the Cartan–Kähler theorem to prove the existence of Lagrangian
submanifolds of complex Euclidean space. Let
ϑ..=dx1∧dy1+dx2∧dy2+···+dxn∧dyn.
Let I be the exterior differential system generated by ϑ on M ..=R2n. Writing
spans of vectors in angle brackets,
Flag Polar equations Characters
E ={0} {0} s =0
0 0
E =h∂ i (cid:10)dy1(cid:11) s =1
1 x1 1
. . .
. . .
. . .
E =h∂ ,∂ ,...,∂ i (cid:10)dy1,dy2,...,dyn(cid:11) s =1
n x1 x2 xn n
The flag predicts
dimM +s +2s +···+ns =2n+1+2+···+n.
1 2 n
The nearby integral elements at a given point of M are parameterized by
dy = adx, which we plug in to ϑ = 0 to see that a can be any symmetric
matrix. So the space of integral elements has dimension
n(n+1) n(n+1)
dimM + =2n+ ,
2 2
correctly predicted. The Cartan–Kähler theorem proves the existence of La-
grangian submanifolds of complex Euclidean space, one (at least) through each
point, tangent to each subspace dy =adx, at least for any symmetric matrix a
close to 0.
1.10 On a complex manifold M, take a Kähler form ϑ and a holomorphic
volume form Ψ, i.e. closed forms expressed in local complex coordinates as
√
−1
ϑ= g dzµ∧dzν¯,
2 µν¯
Ψ =f(z)dz1∧dz2∧···∧dzn,
withf(z)aholomorphicfunctionandg apositivedefiniteself-adjointcomplex
µν¯
matrix offunctions. Prove the existence of special Lagrangian submanifolds, i.e.
integral manifolds of both ϑ and the imaginary part of Ψ.