d¦ + ¦ § ¦ = 0

implies that there is locally (alternatively, on a simply connected cover) a map

g : F3 (M ) ’ O(H)

such that

¦ = g’1 dg.

2.4. THE EQUIVALENCE PROBLEM FOR N ≥ 3 65

Using the structure equations dea = eb •b , this implies

a

d(ea · (g’1 )a ) = 0,

b

so that

¯b

ea = e b ga

for some ¬xed a¬ne frame (¯b ). In particular, the An+1 -valued function e0

e

on F3 (M ) takes as its values precisely the points of a level surface of a non-

degenerate quadratic form, de¬ned by H. Recalling from the ¬rst part of the

proof that x + e0 is constant on An+1, this means that the hypersurface M ,

thought of as the image of the map x : F3 (M ) ’ An+1 , is a constant translate

of a non-degenerate quadric hypersurface. The signature of the quadric is (p, q),

where (p ’ 1, q) is the signature of the ¬rst fundamental form (Hij ).

The case » > 0 instead of » < 0 is quite similar, but M is a quadric of

signature (p, q) when (Hij ) has signature (p, q ’ 1).

The Equivalence Problem for n ≥ 3

2.4

We now consider a contact manifold (M, I) with a closed, almost-classical form

Π = ’θ § (H ij πi § ω(j) ’ Kω). (2.30)

We will shortly specialize to the case in which Π is neo-classical. The coframes

in which Π takes the form (2.30), for some functions H ij and K, constitute a G-

structure as described in Lemma 2.1. The purpose of this section is to describe

a canonical reduction of this G-structure to one carrying a pseudo-connection

satisfying structure equations of a prescribed form, as summarized in (2.47“

2.48), at least in case the matrix (H ij ) is either positive- or negative-de¬nite

everywhere. This application of the equivalence method involves no techniques

beyond those introduced in §2.1, but some of the linear-algebraic computations

are more involved.

We begin by re¬ning our initial G-structure as follows.

Lemma 2.2 Let (M, I) be a contact manifold with almost-classical form Π.

(1) There exist local coframings (θ, ω i , πi) on M such that Π has the form (2.30)

and such that

dθ ≡ ’πi § ωi (mod {I}).

(2) Local coframings as in (1) are the sections of a G0-structure B0 ’ M , where

G0 is the group of matrices of the form (in blocks of size 1, n, n)

«

a 0 0

Ci Ai ,

0 A ∈ GL(n, R), Sij = Sji.

g0 = (2.31)

j

a(A’1 )j

Sik Ak

Di j i

´

66 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

(3) If two local coframings as in (1) are related as

« «

¯

θ θ

ω i = g0 · ω j ,

’1

¯

πi πj

¯

¯ ¯¯ ¯¯

and if Π = ’θ § (H ij πi § ω(j) ’ Kω) = ’θ § (H ij πi § ω(j) ’ K ω) are the

¯

expressions for Π with respect to these coframings, then

¯

a2(det A)A’1 H tA’1 ,

H = (2.32)

¯ ¯

a(det A)(K ’ T r(HS)).

K = (2.33)

Proof. (1) First observe that in any coframing, we may write

dθ ≡ aij πi § πj + bj πj § ωi + cij ωi § ωj (mod {I}).

i

We will deal with each of the three coe¬cient matrices (aij ), (bj ), (cij ) to obtain

i

i

the desired condition dθ ≡ ’ πi § ω .

• The proof of Proposition 2.2 showed for n ≥ 3 that

0 ≡ dθ ≡ aij πi § πj (mod {JΠ }),

which implies aij πi § πj = 0. This followed from calculating 0 = dΠ

modulo {I}.

• From the fact that θ is a contact form, we have

0 = θ § (dθ)n = ±det(bj )θ § ω § π,

i

so that (bj ) is an invertible matrix. Therefore, we may apply the matrix

i

’(bi ) to the 1-forms πj to obtain a new basis in which we have bj = ’δi ,

j ’1 j

i

so that

dθ ≡ ’πi § ωi + cij ωi § ωj (mod {I}).

Note that this coframe change is of the type admitted by Lemma 2.1,

preserving the form (2.30).

• Finally, we can replace πi by πi + cij ωj to have the desired dθ ≡ ’πi § ωi .

This coframe change also preserves the form (2.30).

(2) We already know that any matrix as in Lemma 2.1 will preserve the form

(2.30). We write the action of such a matrix as

±¯

θ = aθ

ωi C θ + Ai ω j

i

¯ = j

j

kj

πi

¯ = Di θ + Sik Aj ω + Bi πj .

2.4. THE EQUIVALENCE PROBLEM FOR n ≥ 3 67

It is easily veri¬ed that the condition dθ ≡ ’πi § ωi implies the analogous

¯

condition dθ ≡ ’¯i § ωi if and only if

π ¯

j j

Bi Ai = aδk ,

k

Sjk = Skj .

This is what we wanted to prove.

¯¯ ¯

(3) These formulae are seen by substituting the formulae for (θ, ωi, πi) into the

equation for the two expressions for Π, and comparing terms. One uses the

following fact from linear algebra: if

ω i ≡ Ai ω j (mod {I}),

¯ j

then

ω(j) ≡ (det A)(A’1 )i ω(i) (mod {I});

¯ j

that is, the coe¬cients of ω(j) in terms of ω(i) are the cofactors of the coe¬cient

¯

matrix of ω in terms of ωj .

i

¯

We can see from (2.32) that the matrix H = (H ij ) transforms under coframe

changes like a bilinear form, up to scaling, and in particular that its absolute

signature is ¬xed at each point of M . To proceed, we have to assume that this

signature is constant throughout M . In particular, we shall from now on assume

that H is positive or negative de¬nite everywhere, and refer to almost-classical

forms Π with this property as de¬nite. Cases of di¬erent constant signature are

of interest, but can be easily reconstructed by the reader in analogy with the

de¬nite case examined below.

Once we assume that the matrix-valued function H on B0 is de¬nite, the

following is an easy consequence of the preceding lemma.

Lemma 2.3 Given a de¬nite, almost-classical Poincar´-Cartan form Π on a

e

contact manifold (M, I), there are 0-adapted local coframings (θ, ω i , πi) for which

Π = ’θ § (δ ij πi § ω(j)),

and these form a G1 -structure B1 ‚ B0 ’ M , where G1 is the group of matrices

g1 of the form (2.31) with

1

a(det A) 2 A ∈ O(n, R),

det A > 0, Sii = 0.

¯ ¯

This follows from imposing the conditions H = H = In , K = K = 0 in the

previous lemma. Unfortunately, it is di¬cult to give a general expression in

coordinates for such a 1-adapted coframing in the classical case, because such

an expression requires that we normalize the Hessian matrix (Lpi pj ). In practice,

however, such a coframing is usually easy to compute.

´

68 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

It is convenient for later purposes to use a di¬erent parameterization of our

group G1. Namely, an arbitrary element will be written as

«

±rn’2 0 0

r’2Ai

g1 = C i ,

0 (2.34)

j

j

Sik Ak ±rn (A’1 )i

Di j

where A = (Ai ) ∈ SO(n, R), r > 0, Sij = Sji, Sii = 0. Also, now that

j

the orthogonal group has appeared, some of the representations occuring in

the sequel are isomorphic to their duals, for which it may be unuseful and

sometimes confusing to maintain the usual summation convention, in which

one only contracts a pair of indices in which one index is raised and the other

lowered. Therefore, we will now sum any index occuring twice in a single term,

regardless of its positions.

We now assume that we have a de¬nite, neo-classical Poincar´-Cartan form

e

Π with associated G1-structure B1 ’ M , and we begin searching for di¬erential

invariants. There are local pseudo-connection 1-forms ρ, γ i , δi, ±i , σij de¬ned so

j

that equations of the following form hold:

« « « «

(n ’ 2)ρ 0 0

θ θ ˜

d ωi = ’ § ω j + „¦i ,

γi ’2ρδj + ±i

i

0

j

nρδi ’ ±j

j

πi πj Πi

δi σij i

where θ, ωi , πi are the tautological 1-forms on B1 , the torsion 2-forms ˜, „¦i, Πi

are semibasic for B1 ’ M , and the psuedo-connection 1-forms satisfy

±i + ±j = 0, σij = σji, σii = 0.

j i

These last conditions mean that the psuedo-connection matrix takes values in

the Lie algebra g1 ‚ gl(2n + 1, R) of G1 .

The psuedo-connection 1-forms are not uniquely determined, and our next

step is to exploit this indeterminacy to try to absorb components of the torsion.

First, we know that dθ ≡ ’πi § ωi (mod {I}). The di¬erence between

˜ = dθ + (n ’ 2)ρ § θ and ’πi § ωi is therefore a semibasic multiple of θ, which

can be absorbed by a semibasic change in ρ. We can therefore simply assume

that

dθ = ’(n ’ 2)ρ § θ ’ πi § ωi ,

or equivalently, ˜ = ’πi § ωi .

Second, our assumption that Π is neo-classical means that the Pfa¬an sys-

tem JΠ = {θ, ωi } is integrable (even up on B1 ). In the structure equation

dωi = ’γ i § θ ’ (’2ρδj + ±i ) § ωj + „¦i ,

i

(2.35)

j

this means that „¦i ≡ 0 (mod {JΠ}). Also, „¦i is semibasic over M , so we can

write

1i