¬ 2· ¬2 ·

0 1 2

ω = ¬ ω · and • = ¬ •0 •1 •2 0 0·

2 2

¬ 3· ¬3 ·

ω •0 0 0 • 3 •3

3 4

ω4 •4 0 0 • 4 •4

0 3 4

are the tautological R5 -valued 1-form and the pseudo-connection form, respec-

tively; note that the condition for • to be g0-valued includes the condition

•0 = • 1 + • 2 = • 3 + • 4 .

0 1 2 3 4

The torsion „ of • is an R5-valued 2-form, semibasic for B0 ’ M and depending

on a choice of pseudo-connection.

Returning to the general situation of a G-structure B ’ M , our goal is to

understand how di¬erent choices of pseudo-connection in (2.8) yield di¬erent

torsion forms. We will use this to restrict attention to those pseudo-connections

whose torsion is in some normal form.

The linear-algebraic machinery for this is as follows. Associated to the linear

Lie algebra g ‚ gl(n, R) is a map of G-modules

2

δ : g — (Rn )— ’ Rn — (Rn )— ,

de¬ned as the restriction to

g — (Rn)— ‚ (Rn — (Rn )— ) — (Rn)— (2.9)

of the surjective skew-symmetrization map

2

Rn — (Rn)— — (Rn )— ’ Rn — (Rn)— .

The cokernel of δ

def 2

(Rn )— )/δ(g — (Rn)— )

H 0,1(g) = (Rn — (2.10)

´

44 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

is one of the Spencer cohomology groups of g ‚ gl(n, R). Note that to each b ∈ B

∼

is associated an isomorphism Tπ(b) M ’ Rn , and consequently an identi¬cation

of semibasic 1-forms at b ∈ B with (Rn )— . Now, given a pseudo-connection in

the G-structure, the semibasic Rn -valued torsion 2-form ( 1 Tjk ωj § ωk ) at b ∈ B

i

2

can be identi¬ed with an element „b ∈ Rn — Λ2(Rn )— . Similarly, a permissible

change at b ∈ B of the pseudo-connection”that is, a semibasic g-valued 1-

form”can be identi¬ed with an element •b ∈ g — (Rn )— . Under these two

identi¬cations, the map δ associates to a change •b the corresponding change

in the torsion •b § ωb, where in this expression we have contracted the middle

factor of (Rn )— in •b (see (2.9)) with the values of the Rn-valued 1-form ωb.

Therefore, di¬erent choices of pseudo-connection yield torsion maps di¬ering by

elements of Im(δ), so what is determined by the G-structure alone, independent

of a choice of pseudo-connection, is a map „ : B ’ H 0,1(g), called the intrinsic

¯

torsion of B ’ M .

This suggests a major step in the equivalence method, called absorption of

torsion, which one implements by choosing a (vector space) splitting of the

projection

2

(Rn )— ’ H 0,1(g) ’ 0.

Rn — (2.11)

As there may be no G-equivariant splitting, one is merely choosing some vector

2

subspace T ‚ Rn — (Rn )— which complements the kernel δ(g—(Rn )— ). Fixing

a choice of T , it holds by construction that any G-structure B ’ M locally has

pseudo-connections whose torsion at each b ∈ B corresponds to a tensor lying

in T .

We will see from our example of hyperbolic Monge-Ampere systems that this

is not as complicated as it may seem. Denote the semibasic 2-form components

of the R5 -valued torsion by « 0

„

¬ „1 ·

¬ ·

„ = ¬ „2 · .

¬ 3·

„

„4

We know from the condition (2.4) in the de¬nition of 0-adapted that

def

„ 0 = dω0 + •0 § ω0 = ω1 § ω2 + ω3 § ω4 + σ § ω0

0

for some semibasic 1-form σ. We may now replace •0 by •0 ’ σ in our pseudo-

0 0

0

connection, eliminating the term σ § ω from the torsion. We then rename this

altered pseudo-connection entry again as •0 ; to keep the pseudo-connection g0 -

0

valued, we have to make a similar change in •1 + •2 and •3 + •4 . What we

1 2 3 4

have just shown is that given an arbitrary pseudo-connection in a G0 -structure

B0 ’ M , there is another pseudo-connection whose torsion satis¬es (using

obvious coordinates on Rn — 2(Rn )— ) T0a = Ta0 = 0. By choosing this latter

0 0

pseudo-connection, we are absorbing the corresponding torsion components into

2.1. THE EQUIVALENCE PROBLEM FOR n = 2 45

•. Furthermore, the fact that our G0-structure is not arbitrary, but comes from

a hyperbolic Monge-Ampere system, gave us the additional information that

0 0 0

T12 = T34 = 1, and all other independent Tij = 0. Note incidentally that

our decision to use a pseudo-connection giving σ = 0 determines •0 uniquely,

0

up to addition of multiples of ω 0 ; this uniqueness applies also to •1 + •2 and

1 2

•3 +•4 . The e¬ort to uniquely determine pseudo-connection forms should guide

3 4

the choices one makes in the equivalence method.

Other torsion terms may be absorbed using similar methods. Using the

index range 1 ¤ i, j, k ¤ 4, we write

„ i = Tj0ωj § ω0 + 1 Tjk ωj § ωk

i i

2

for functions Tj0 and Tjk = ’Tkj . First, by altering the nilpotent part •i , we

i i i

0

can arrange that all Tj0 = 0. Second, by altering the o¬-diagonal terms •2, •2 ,

i 1

1

•4 , •3 , we can arrange that

3 4

1 2 3 4

T2j = T1j = T4j = T3j = 0.

Third, by altering the traceless diagonal parts •1 ’ •2 and •3 ’ •4 , we can

1 2 3 4

arrange that

1 2 1 2 3 4 3 4

T13 = T23, T14 = T24, T13 = T14, T23 = T24.

We summarize this by renaming

„1 (V3 ω3 + V4 ω4 ) § ω1 + U 1 ω3 § ω4 ,

=

„2 (V3 ω3 + V4 ω4 ) § ω2 + U 2 ω3 § ω4 ,

=

„3 (V1 ω1 + V2 ω2 ) § ω3 + U 3 ω1 § ω2 ,

=

„4 (V1 ω1 + V2 ω2 ) § ω4 + U 4 ω1 § ω2 ,

=

a

for 8 torsion functions Ui , Vi on B0 . The collection of torsion tensors (Tbc )

0 0

taking this form, and satisfying T0a = Ta0 = 0, constitutes the splitting of

(2.11) given in the general discussion, to which we will return shortly.

At this point, we can uncover more consequences of the fact that we are

dealing not with an arbitrary G0-structure on a 5-manifold, but a special one

induced by a hyperbolic Monge-Ampere system. We already found as one con-

sequence the fact that

„ 0 ≡ ω1 § ω2 + ω3 § ω4 (mod {ω0 }),

which has nothing to do with our choices in absorbing torsion; absorbing torsion

allowed us to render this congruence into an equality. Similarly, we now obtain

pointwise relations among other torsion coe¬cients by computing, modulo {I}

´

46 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

(which in this case means ignoring all ω 0 terms after di¬erentiating),

d(dω0)

≡

0

•0 § (ω1 § ω2 + ω3 § ω4 )

≡ 0

+((’•1 + V3 ω3 + V4 ω4 ) § ω1 + U 1 ω3 § ω4 ) § ω2

1

’ω § ((’•2 + V3 ω3 + V4ω4 ) § ω2 + U 2 ω3 § ω4 )

1

2

+((’•3 + V1 ω1 + V2 ω2 ) § ω3 + U 3 ω1 § ω2 ) § ω4

3

’ω3 § ((’•4 + V1 ω1 + V2ω2 ) § ω4 + U 4 ω1 § ω2 )

4

(U 1 + 2V2 )ω2 § ω3 § ω4 ’ (U 2 ’ 2V1)ω1 § ω3 § ω4

≡

+(U 3 + 2V4 )ω1 § ω2 § ω4 ’ (U 4 ’ 2V3 )ω1 § ω2 § ω3 ,

so that

U 1 = ’2V2 , U 2 = 2V1 , U 3 = ’2V4 , U 4 = 2V3.

These are pointwise linear-algebraic relation among our 8 torsion functions.

In the general study of G-structures B ’ M , we now have to consider the

group action in more detail. Speci¬cally, H 0,1(g) is the cokernel of a map of G-

modules, so it inherits a G-action as well, and it is easy to see that the intrinsic

torsion „ : B ’ H 0,1(g) is equivariant for this action. Therefore, there is an

¯

induced map

[¯] : M ’ H 0,1(g)/G,

„

which is an invariant of the equivalence class of the G-structure B ’ M ; that is,

under a di¬eomorphism M1 ’ M2 inducing an equivalence of G-structures, [„2]

must pull back to [„1]. Now, H 0,1(g)/G typically has a complicated topology,

and is rarely a manifold. However, in many cases of interest one can ¬nd a

slice W ‚ H 0,1(g), a submanifold whose points all have the same stabilizer

G1 ‚ G, and which is a cross-section of the orbits which W itself intersects. If

the intrinsic torsion „ : B ’ H 0,1(g) of a G-structure takes values in a union of

¯

orbits represented by such a slice, then the set

def

B1 = „ ’1(W )

¯

is a smooth principal subbundle of B ’ M having structure group G1 ‚ G.

The process of reducing to a subbundle de¬ned as the locus where intrinsic

torsion lies in a slice is called normalizing the torsion. If G1 is a proper subgroup

of G, then we can essentially start the process over, starting with an arbitrary

pseudo-connection, absorbing torsion, and so on. Typically, one inherits from

B ’ M some information about the torsion of the subbundle B1 ’ M , because

the original structure equations restrict to the submanifold B1 ‚ B. We will

see an example of this below.

In practice, one typically studies the G-action on H 0,1(g) by transporting it

2

(Rn)— . If T is not an invariant

to the representing vector space T ‚ Rn —

2

subspace of Rn — (Rn)— , then typically G will act by a¬ne-linear motions on

2.1. THE EQUIVALENCE PROBLEM FOR n = 2 47

T . This is the case in the next step of our equivalence problem for hyperbolic

Monge-Ampere systems.

We have represented the intrinsic torsion of a G0 -structure B0 ’ M corre-

sponding to a hyperbolic Monge-Ampere system by 4 independent functions on

B; that is, our torsion takes values in a 4-dimensional subspace of the lift T of

H 0,1(g0 ). The next step is to determine how the independent torsion functions

vary along the ¬bers of B0 ’ M . This will be expressed in¬nitesimally, in an

equation for the exterior derivative of the torsion functions, modulo the space

of forms that are semibasic for B0 ’ M ; the expressions will be in terms of

the pseudo-connection forms which parallelize the ¬bers. They are obtained as

follows.

We ¬rst consider the equations for dω 1 , dω2 . Taking the exterior derivative

of each, modulo the algebraic ideal {ω 0, ω1 , ω2}, yields equivalences of 3-forms

that do not involve derivatives of any psuedo-connection forms, but do involve

dU 1, dU 2. From each of these can be factored the 2-form ω 3 § ω4 , yielding a

pair of equivalences modulo {ω 0 , . . . , ω4}, expressible in matrix form as

U1 •1 •1 •1 U1 U1

’ •0 ·

0 1 2

0≡d ·

+ + .

U2 •2 •2 •2 U2 0

U2

0 1 2

A similar procedure applied to the equations for dω 3 , dω4 yields the pair

U3 •3 •3 •3 U3 U3

•0

0 3 4

0≡d · ’ ·

+ + .

U4 •4 •4 •4 U4 U4

0

0 3 4