2

2.4. THE EQUIVALENCE PROBLEM FOR n ≥ 3 69

Now, adding semibasic 1-forms to γ i allows us to preserve the equation (2.35)

while also making (2.36) an equality, and not merely a congruence. A little

linear algebra shows that there is a unique linear combination of the ω i that can

be added to ±i , preserving ±i + ±j = 0, to absorb the term 1 Pjkωj § ωk . This

i

j j i 2

leaves us only with

„¦i = T ijkπj § ωk .

As in the elimination of the Pjk , we can add a combination of the πi to ±i to

i

j

arrange

T ijk = T kji.

To investigate the third torsion term Πi , we use an alternate derivation of

the equation for dπi. Namely, we di¬erentiate the equation

dθ = ’(n ’ 2)ρ § θ ’ πi § ωi ,

and take the result only modulo {I} to avoid the unknown quantity dρ. This

eventually yields

0 ≡ ’(Πk ’ T ijkπi § πj ) § ωk (mod {I}).

As before, multiples of θ may be absorbed by rede¬ning δi , so that we can

assume this congruence is an equality. Reasoning similar to that which proves

the Cartan lemma gives

Πk ’ T ijkπi § πj = νkl § ωl

for some semibasic 1-forms νkl = νlk . Now, most of these forms νkl can be

subtracted from the psuedo-connection forms σkl , simplifying the torsion; but

the condition σii = 0 prevents us from completely absorbing them. Instead, the

trace remains, and we have

Πk = δkl ν § ωl + T ijkπi § πj .

We can learn more about ν using the integrability condition dΠ = 0, taken

modulo terms quadratic in the πi :

0 = dΠ ≡ nθ § ν § ω.

A consequence is that ν ≡ 0 (mod {θ, ω i }); in other words, ν has no πi-terms,

and may be written (using again a change in δi ) as

Ni ω i .

ν=

Then replacing σij by

+ δjk Ni ’ n δij Nk )ωk

n 2

σij + n+2 (δik Nj

yields new psuedo-connection forms, for which the third torsion term is simply

Πk = T ijkπi § πj .

´

70 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

This completes the major step of absorbing torsion by altering the pseudo-

connection.

Before proceeding to the next major step, we look for linear-algebraic con-

ditions on the torsion which may simplify later calculations. In particular, we

made only very coarse use of dΠ = 0 above. Now we compute more carefully

0 = dΠ = ’θ § (2T ijk + δ ij T lkl )πk § πj § ω(i),

so we must have

2T ijk + δ ij T lkl = 2T ikj + δ ik T ljl . (2.37)

The next major step is a reduction of our G1-structure. We will examine

def

the variation of the functions T j = T iji along ¬bers of B1 ’ M , and observe

that the zero-locus {T j = 0} de¬nes a G2-structure for a certain codimension-n

subgroup G2 ‚ G1 .

As usual, the variation of T j will be described in¬nitesimally. To study dT j

without knowledge of the traceless part of dT ijk , we exploit the exterior algebra,

writing

d(θ § ω1 § · · · § ωn ) = ((n + 2)ρ + T k πk ) § θ § ω. (2.38)

We will di¬erentiate this for information about dT k , but in doing so we will need

information about dρ as well. Fortunately, this is available by di¬erentiating

the ¬rst structure equation

dθ = ’(n ’ 2)ρ § θ ’ πk § ωk ,

yielding

(n ’ 2)dρ ≡ γ k § πk (mod {θ, ωi }).

Now we return to di¬erentiating (2.38) and eventually ¬nd

dT k ≡ ’ n+2 γ k + (nδj ρ ’ ±k )T j

k

(mod {θ, ωi , πi}).

j

n’2

This means that along ¬bers of B1 ’ M , the vector-valued function T (u) =

(T j (u)), u ∈ B1 , is orthogonally rotated (in¬nitesimally, by ±k ), scaled (by ρ),

j

and translated (by γ i ). In fact, for g1 ∈ G1 as in (2.34),

T (u · g1) = ±r2A’1 (rn’2T (u) ’ n+2

n’2 C).

Now the set

def

B2 = {u ∈ B1 : T (u) = 0} ‚ B1

is a G2-subbundle of B1 ’ M , where G2 consists of matrices as in (2.34) with

T i = 0.

On the submanifold B2 ‚ B1 , we have from (2.37) the symmetry

T ijk = T kji = T ikj .

2.4. THE EQUIVALENCE PROBLEM FOR n ≥ 3 71

As a consequence, the torsion Πk restricts to

Πk = T ijkπj § πk = 0.

The previous structure equations continue to hold, but the forms γ i |B2

should not be regarded as part of the psuedo-connection, as they are now semiba-

sic over M . We therefore write

« « « «

(n ’ 2)ρ 0 0

θ θ ˜

d ωi = ’ § ω j + „¦i ,

’2ρδj + ±i

i

0 0

j

nρδi ’ ±j

j

πi πj Πi

δi σij i

(2.39)

where still

±i + ±j = 0, σij = σji, σii = 0, (2.40)

j i

and now

±

˜ = ’πi § ωi ,

„¦i = ’(Sj ωj + U ij πj ) § θ + T ijk πj § ωk ,

i

(2.41)

Πi = 0.

Here we have denoted γ i ≡ Sj ωj + U ij πj (mod {I}). Also, we still have

i

T ijk = T kji = T ikj , T iik = 0. (2.42)

Notice that we can alter ±i and ρ to assume that

j

j

i i

Sj = S i , Si = 0, (2.43)

where we also have to add combinations of ω i to δi to preserve Πi = 0. In fact,

these assumptions uniquely determine ±i and ρ, although δi and σij still admit

j

some ambiguity.

Equations (2.39“2.43) summarize the results of the equivalence method car-

ried out to this point. We have uncovered the primary di¬erential invariants of

a de¬nite neo-classical Poincar´-Cartan form: they are the functions T ijk , Sj i

e

and U ij . Their properties are central in what follows.

For example, note that the rank-n Pfa¬an system {ω i } on B2 is invariant

under the action of the structure group G2, and therefore it is the pullback of a

Pfa¬an system (also to be denoted {ω i }) down on M . Testing its integrability,

we ¬nd

dωi ≡ ’U ij πj § θ (mod {ωi }). (2.44)

We will see shortly that the matrix-valued function (U ij ) varies along the ¬bers

of B2 ’ M by a linear representation of G2, so that it is plausible to ask

about those Poincar´-Cartan forms for which U ij = 0; (2.44) shows that this

e

´

72 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

is equivalent to the integrability of {ω i }. In this case, in addition to the local

¬bration M ’ Q whose ¬bers are leaves of JΠ , we have

Qn+1 ’ N n,

where N is the locally-de¬ned n-dimensional “leaf space” for {ω i }. Coordinates

on N ”equivalently, functions on M whose di¬erentials lie in {ω i}”may be

thought of as “preferred independent variables” for the contact-equivalence class

of our Euler-Lagrange equation, canonical in the sense that every symmetry of

M preserving the Poincar´-Cartan form preserves the ¬bration M ’ N and

e

therefore acts on N . Note that even if an (M, Π) satisfying U ij = 0 came to us

from a classical Lagrangian with independent variables (xi ), we need not have

{ωi } = {dxi}.

This is not to say that the case U ij = 0 is uninteresting. In the next section,

we will see an important family of examples from Riemannian geometry with

U ij = »δj . To obtain preliminary information about U ij in a manner that will

i

not require much knowledge of Sj or T ijk , we start with the equation

i

d(ω1 § · · · § ωn ) = 2nρ § (ω1 § · · · § ωn ) + U ij θ § πj § ω(i) . (2.45)

We will di¬erentiate again, but we need more re¬ned information about dρ; this

is obtained from

0 = d2θ = ’((n ’ 2)dρ + δi § ωi + πi § γ i ) § θ.

Keep in mind that γ i = Sj ωj + U ij πj (mod {I}) on this reduced bundle. We

i

can now write

(n ’ 2)dρ + δi § ωi + πi § γ i = „ § θ (2.46)

for some unknown 1-form „ . Returning to the derivative of (2.45), we ¬nd

§ U ij πj ) § ω + U ij πi § πj § ω

2n

0≡ (mod {I}).

n’2 (’πi

This implies that U ij πi § πj § ω = 0, so that we have

U ij = U ji .

We will need an even more re¬ned version of the equation (2.46) for dρ.

In the preceding paragraph, we substituted that equation into the equation for

0 ≡ d2(ω1 § · · · § ωn) (mod {I}). Now, we substitute it instead into

d2(ω1 § · · · § ωn ) (mod {π1, . . . , πn})

0≡

’ U ij σij § θ § ω1 § · · · § ωn.

2n

≡ n’2 „

2n

This means that n’2 „ ’ U ij σij lies in {θ, ωi , πi}. Recall that also γ i = Sj ωj +

i

U ij πj + V i θ for some functions V i , and we can put this back into (2.46) to

¬nally obtain

(n ’ 2)dρ = ’δi § ωi ’ Sj πi § ωj +

i

U ij σij § θ + (si ωi ’ ti πi) § θ,

n’2

2n

2.4. THE EQUIVALENCE PROBLEM FOR n ≥ 3 73

for some functions si , ti . Furthermore, we can replace each δi by δi ’ si θ,

preserving previous equations, to assume that si = 0. This gives

(n ’ 2)dρ = ’δi § ωi ’ Sj πi § ωj +

i

U ij σij § θ ’ ti πi § θ,

n’2

2n

which will be used in later sections.

The last formulae that we will need are those for the transformation rules for

T , U ij , Sj along ¬bers of B ’ M . These are obtained by computations quite

ijk i

similar to those carried out above, and we only state the results here, which are:

• dT ijk ≡ nρT ijk ’ ±i T ljk ’ ±j T ilk ’ ±k T ijl ,

l l l