hypersurfaces in Riemannian manifolds.

In the next chapter, we will specialize to the study of those neo-classical

Poincar´-Cartan forms whose primary di¬erential invariants all vanish. These

e

correspond to interesting variational problems arising in conformal geometry.

We begin with a few elementary notions used in the method of equivalence.

On a manifold M of dimension n, a coframe at a point x ∈ M is a linear

isomorphism

∼

ux : Tx M ’’ Rn .

—

This is equivalent to a choice of basis for the cotangent space Tx M , and we will

not maintain any distinction between these two notions. The set of all coframes

for M has the structure of a principal GL(n, R)-bundle π : F(M ) ’ M , with

right-action

def

ux · g = g’1 ux , g ∈ GL(n, R),

where the right-hand side denotes composition of ux with multiplication by g ’1 .

A local section of π : F(M ) ’ M is called a coframing, or coframe ¬eld. On

the total space F(M ), there is an Rn-valued tautological 1-form ω, given at

u ∈ F(M ) by

ωu (v) = u(π— v) ∈ Rn , v ∈ Tu F(M ). (2.1)

The n components ω i of this Rn-valued 1-form give a global basis for the semiba-

sic 1-forms of F(M ) ’ M .

In terms of coordinates x = (x1, . . . , xn) on M , there is a trivialization

M — GL(n, R) ∼ F(M ) given by

=

(x, g) ” (x, g’1dx),

where on the right-hand side, dx is a column of 1-forms regarded as a coframe

at x, and g’1 dx is the composition of that coframe with multiplication by

g’1 ∈ GL(n, R). In this trivialization, we can express the tautological 1-form

as

ω = g’1 dx,

where again the right-hand side represents the product of a GL(n, R)-valued

¬ber coordinate and an Rn-valued semibasic 1-form.

The geometric setting of the equivalence method is the following.

De¬nition 2.1 Let G ‚ GL(n, R) be a subgroup. A G-structure on the n-

manifold M is a principal subbundle of the coframe bundle F(M ) ’ M , having

structure group G.

We will associate to a hyperbolic Monge-Ampere system (to be de¬ned, in case

n = 2), or to a neo-classical Poincar´-Cartan form (in case n ≥ 3), a succession

e

of G-structures on the contact manifold M , which carry increasingly detailed

information about the geometry of the system or form, respectively.

2.1. THE EQUIVALENCE PROBLEM FOR N = 2 39

2.1 The Equivalence Problem for n = 2

In this section, we will study the equivalence problem for certain Monge-Ampere

systems on contact manifolds of dimension 5. We will give criteria in terms of

the di¬erential invariants thus obtained for a given system to be locally equiv-

alent the system associated to the linear homogeneous wave equation. We will

also give the weaker criteria for a given system to be locally equivalent to an

Euler-Lagrange system, as in the previously discussed inverse problem. Unless

otherwise noted, we use the index ranges 0 ¤ a, b, c ¤ 4, 1 ¤ i, j, k ¤ 4.

We assume given a 5-dimensional contact manifold (M, I) and a Monge-

Ampere system E, locally algebraically generated as

E = {θ, dθ, Ψ},

where 0 = θ ∈ “(I) is a contact form, and Ψ ∈ „¦2(M ) is some 2-form. As

noted previously, E determines I and I. We assume that Ψx ∈ Ix for all

/

x ∈ M . Recall from the discussion in §1.2.3 that given E, the generator Ψ may

be uniquely chosen modulo {I} (and modulo multiplication by functions) by

the condition of primitivity; that is, we may assume

dθ § Ψ ≡ 0 (mod {I}).

The assumption Ψx ∈ Ix means that this primitive form is non-zero everywhere.

/

We do not necessarily assume that E is Euler-Lagrange.

On the contact manifold M , one can locally ¬nd a coframing · = (· a ) such

that

·0 ∈ “(I),

(2.2)

d·0 ≡ ·1 § ·2 + ·3 § ·4 (mod {I}).

1

Then we can write Ψ ≡ 2 bij ·i § ·j (mod {I}), where the functions bij depend

on the choice of coframing and on the choice of Ψ. The assumption that Ψ is

primitive means that in terms of a coframing satisfying (2.2),

b12 + b34 = 0.

We now ask what further conditions may be imposed on the coframing · = (· a )

while preserving (2.2).

To investigate this, we ¬rst consider changes of coframe that ¬x · 0 ; we will

later take into account non-trivial rescalings of · 0 . In this case, an element

of GL(5, R) preserves the condition (2.2) if and only if it acts as a ¬berwise

sympletic transformation, modulo the contact line bundle I. Working modulo

I, we can split

(T — M/I) ∼ (R · d·0) • P 2(T — M/I),

2

=

where P 2 (T — M/I) is the 5-dimensional space of 2-forms that are primitive with

respect to the symplectic structure on I ⊥ induced by d·0. The key observation

is that the action of the symplectic group Sp(2, R) on P 2(R4 ) is equivalent to

´

40 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

the standard action of the group SO(3, 2) on R5. This is because the symmetric

bilinear form ·, · on P 2(T — M/I) de¬ned by

ψ1 § ψ2 = ψ1 , ψ2 (d·0 )2

has signature (3, 2) and symmetry group Sp(2, R). Therefore, the orbit de-

composition of the space of primitive forms Ψ modulo {I} under admissible

changes of coframe will be a re¬nement of the standard orbit decomposition

under SO(3, 2).

To incorporate rescaling of · 0 into our admissible changes of coframe, note

that a rescaling of · 0 requires via (2.2) the same rescaling of the symplectic

form ·1 § ·2 + ·3 § ·4, so we should actually allow changes by elements of

GL(5, R) inducing the standard action of CSp(2, R); this is the group that

preserves the standard symplectic form up to scale. This in turn corresponds

to the split-signature conformal group CO(3, 2), which acts on R5 with three

non-zero orbits: a negative space, a null space, and a positive space.

The three orbits of this representation correspond to three types of Monge-

Ampere systems:

• If Ψ § Ψ is a negative multiple of d· 0 § d·0, then the local coframing ·

may be chosen so that in addition to (2.2),

Ψ ≡ ·1 § ·2 ’ ·3 § ·4 (mod {I});

for a classical variational problem, this occurs when the Euler-Lagrange

PDE is hyperbolic.

• If Ψ § Ψ = 0, then · may be chosen so that

Ψ ≡ ·1 § ·3 (mod {I});

for a classical variational problem, this occurs when the Euler-Lagrange

PDE is parabolic.

• If Ψ § Ψ is a positive multiple of d· 0 § d·0 , then · may be chosen so that

Ψ ≡ ·1 § ·4 ’ ·3 § ·2 (mod {I});

for a classical variational problem, this occurs when the Euler-Lagrange

PDE is elliptic.

The equivalence problem for elliptic Monge-Ampere systems in case n = 2 devel-

ops in analogy with that for hyperbolic systems; we will present the hyperbolic

case. The conclusion will be:

Associated to a hyperbolic Monge-Ampere system (M 5, E) is a canon-

ical subbundle B1 ’ M of the coframe bundle of M carrying a pair

of 2 — 2-matrix-valued functions S1 and S2 , involving up to second

derivatives of the given system. (M, E) is locally of Euler-Lagrange

type if and only if S2 vanishes identically, while it is equivalent to

the system associated to the homogeneous wave equation zxy = 0 if

and only if S1 and S2 both vanish identically.

2.1. THE EQUIVALENCE PROBLEM FOR n = 2 41

An example of a hyperbolic Monge-Ampere system, to be studied in more detail

in Chapter 4, is the linear Weingarten system for surfaces in E3 with Gauss

curvature K = ’1.

To begin, assume that (M 5 , E) is a hyperbolic Monge-Ampere system. A

coframing · = (· a ) of M is said to be 0-adapted to E if

E = {·0, ·1 § ·2 + ·3 § ·4 , ·1 § ·2 ’ ·3 § ·4} (2.3)

and also

d·0 ≡ ·1 § ·2 + ·3 § ·4 (mod {I}). (2.4)

According to the following proposition, a hyperbolic Monge-Ampere system is

equivalent to a certain type of G-structure, and it is the latter to which the

equivalence method directly applies.

Proposition 2.1 The 0-adapted coframings for a hyperbolic Monge-Ampere

system (M 5, E) are the sections of a G0-structure on M , where G0 ‚ GL(5, R)

is the (disconnected) subgroup generated by all matrices of the form (displayed

in blocks of size 1, 2, 2)

«

a00

g0 = C A 0 , (2.5)

D0B

with a = det(A) = det(B) = 0, along with the matrix

«

100

J = 0 0 I2 . (2.6)

0 I2 0

Proof. The content of this proposition is that any two 0-adapted coframes

di¬er by multiplication by an element of G0 . To see why this is so, note that

the 2-forms

·1 § ·2 + ·3 § ·4 and ·1 § ·2 ’ ·3 § ·4

have, up to scaling, exactly 2 decomposable linear combinations, · 1 § ·2 and

·3 § ·4 . These must be either preserved or exchanged by any change of coframe

preserving their span modulo {I}, and this accounts for both the block form

(2.5) and the matrix J. The condition on determinants then corresponds to

(2.4).

Although not every G0-structure on a 5-manifold M is induced by a hyper-

bolic Monge-Ampere system E, it is easy to see that those that do determine

E uniquely. We therefore make a digression to describe the ¬rst steps of the

equivalence method, by which one investigates the local geometry of a general

G-structure. This will be followed by application to the case at hand of a G0 -

structure, then a digression on the next general steps, and application to the

´

42 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

case at hand, and so on. One major step, that of prolongation, will not ap-

pear in this chapter but will be discussed in the study of conformal geometry in

Chapter 3.

Fix a subgroup G ‚ GL(n, R). Two G-structures Bi ’ Mi , i = 1, 2, are

equivalent if there is a di¬eomorphism M1 ’ M2 such that under the induced

isomorphism of principal coframe bundles F(M1 ) ’ F(M2 ), the subbundle

B1 ‚ F(M1 ) is mapped to B2 ‚ F(M2 ). One is typically interested only

in those properties of a G-structure which are preserved under this notion of

equivalence. For instance, if one has a pair of 5-manifolds with hyperbolic

Monge-Ampere systems, then a di¬eomorphism of the 5-manifolds carries one

of these systems to the other if and only if it induces an equivalence of the

associated G0-structures.

It is easy to see that a di¬eomorphism F : B1 ’ B2 between the total spaces

of two G-structures Bi ’ Mi is an equivalence in the above sense if and only if

F — (ω2 ) = ω1 , where ωi is the restriction of the tautological Rn-valued form (2.1)

on F(Mi ) ⊇ Bi . The ¬rst step in investigating the geometry of a G-structure

B ’ M is therefore to understand the local behavior of this tautological form.

To do this, we seek an expression for its exterior derivative, and to understand

what such an expression should look like, we proceed as follows.

Consider a local trivialization B ∼ M — G, induced by a choice of section ·

=

of B ’ M whose image is identi¬ed with M — {e} ‚ M — G. The section · is

in particular an Rn -valued 1-form on M , and the tautological 1-form is

ω = g’1 · ∈ „¦1(B) — Rn.

The exterior derivative of this equation is

dω = ’g’1 dg § ω + g’1 d·. (2.7)

Note that the last term in this equation is semibasic for B ’ M , and that the

matrix 1-form g ’1 dg takes values in the Lie algebra g of G. Of course, these

pieces g’1 d· and g’1 dg each depend on the choice of trivialization. To better

understand the pointwise linear algebra of (2.7), we introduce the following

notion.

De¬nition 2.2 A pseudo-connection in the G-structure B ’ M is a g-valued

1-form on B whose restriction to the ¬ber tangent spaces Vb ‚ Tb B equals the

identi¬cation Vb ∼ g induced by the right G-action on B.

=

This di¬ers from the de¬nition of a connection in the principal bundle B ’ M

by omission of an equivariance requirement. In terms of our trivialization above,

a pseudo-connection on M — G is any g-valued 1-form of the form

g’1 dg + (semibasic g-valued 1-form);

in particular, every G-structure carries a pseudo-connection. A consequence of

(2.7) is that any pseudo-connection • ∈ „¦1 (B) — g satis¬es a structure equation

2.1. THE EQUIVALENCE PROBLEM FOR n = 2 43

that is fundamental for the equivalence method:

dω = ’• § ω + „, (2.8)

where „ = ( 2 Tjk ωj §ωk ) is a semibasic Rn-valued 2-form on B, called the torsion

1i

of the pseudo-connection •. It is natural to consider exactly how a di¬erent

choice of pseudo-connection”remember that any two di¬er by an arbitrary

semibasic g-valued 1-form”yields a di¬erent torsion form. We will pursue this

after considering the situation for our hyperbolic Monge-Ampere systems.

Let B0 ‚ F(M ) be the G0-bundle of 0-adapted coframes for a hyperbolic

Monge-Ampere system E. A local section · corresponds to an R5 -valued 1-form

(·a ) satisfying (2.3, 2.4). In terms of the trivialization B0 ∼ M — G0 induced by

=

’1

5

·, the tautological R -valued 1-form is ω = g0 ·. Locally (over neighborhoods

in M ), there is a structure equation (2.8), in which

« 0 «0

ω •0 0 0 0 0