ńņš. 14 |

d d

F p ā„¦0 ā’ā’ F p ā„¦1 ā’ā’ Ā· Ā· Ā·

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56 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

is locally exact for each p. Now, any almost-classical form Ī lies in F n ā„¦1; so

not only is the closed form Ī locally equal to dĪ for some Ī ā ā„¦n (M ), we

can actually choose Ī to lie in F n ā„¦0. In other words, we can locally ļ¬nd a

Lagrangian Ī of the form

Ī = L0 (x, z, p)dx + Li (x, z, p)dz ā§ dx(i)

for some functions L0 , Li . This may be rewritten as

Ī = (L0 + pi Li )dx + Īø ā§ (Li dx(i)),

and then the condition Īø ā§ dĪ = 0 (recall that this was part of the construction

of the PoincarĀ“-Cartan form associated to any class in H n (ā„¦ā— /I)) gives the

e

relation

ā‚L

Li (x, z, p) = (x, z, p).

ā‚pi

This is exactly the condition for Ī to locally be a classical Lagrangian.

Returning to the geometry associated to a neo-classical PoincarĀ“-Cartan e

form Ī , we have found (or in case n = 3, postulated) an integrable Pfaļ¬an sys-

tem JĪ which is invariant under contact transformations preserving Ī . Locally

in M , the induced foliation has a smooth āleaf-spaceā Q of dimension n + 1,

and there is a smooth submersion q : M ā’ Q whose ļ¬bers are n-dimensional

integral manifolds of JĪ . On such a neighborhood, the foliation will be called

simple, and as we are only going to consider the local geometry of Ī in this

section, we assume that the foliation is simple on all of M . We may restrict to

smaller neighborhoods as needed in the following.

To explore the geometry of the situation, we ask what the data (M 2n+1, I, Ī )

look like from the point of view of Qn+1. The ļ¬rst observation is that we can

locally identify M , as a contact manifold, with the standard contact manifold

Gn (T Q), the Grassmannian bundle parameterizing n-dimensional subspaces of

ļ¬bers of T Q. This is easily seen in coordinates as follows. If, as in the preceding

proof, we integrate JĪ as

JĪ = {dz, dxi}

for some local functions z, xi on M , then the same functions z, xi may be re-

garded as coordinates on Q. With the assumption that Īø ā {dxi} (on M , again),

/

i

we must have dz ā’ pidx ā Ī“(I) for some local functions pi on M , which by the

non-degeneracy condition for I make (xi , z, pi) local coordinates on M . These

pi can also thought of as local ļ¬ber coordinates for M ā’ Q, and we can map

M ā’ Gn(T Q) by

(xi, z, pi) ā’ ((xi , z); {dz ā’ pi dxi}ā„ ).

The latter notation refers to a hyperplane in the tangent space of Q at (xi, z).

Under this map, the standard contact system on Gn(T Q) evidently pulls back

to I, so we have a local contact diļ¬eomorphism commuting with projections to

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2.2. NEO-CLASSICAL POINCARE-CARTAN FORMS 57

Q. Every point transformation of Q prolongs to give a contact transformation

of Gn(T Q), hence of M as well. Conversely, every contact transformation of

M that preserves Ī is the prolongation of a point transformation of Q, be-

cause the foliation by integral manifolds of JĪ deļ¬ning Q is associated to Ī

in a contact-invariant manner.4 In this sense, studying the geometry of a neo-

classical PoincarĀ“-Cartan form (in case n ā„ 3) under contact transformations is

e

locally no diļ¬erent than studying the geometry of an equivalence class of classi-

cal non-degenerate ļ¬rst-order scalar Lagrangians under point transformations.

We have now interpreted (M, I) as a natural object in terms of Q, but our

real interest lies in Ī . What kind of geometry does Ī deļ¬ne in terms of Q? We

will answer this question in terms of the following notion.

Deļ¬nition 2.5 A Lagrangian potential for a neo-classical PoincarĀ“-Cartan

e

n0

form Ī on M is an n-form Ī ā F ā„¦ (that is, Ī is semibasic for M ā’ Q) such

that dĪ = Ī .

We saw in the proof of Proposition 2.3 that locally a Lagrangian potential Ī

exists. Such Ī are not unique, but are determined only up to addition of closed

forms in F n ā„¦0. It will be important below to note that a closed form in F n ā„¦0

must actually be basic for M ā’ Q; that is, it must be locally the pull-back of a

(closed) n-form on Q. In particular, the diļ¬erence between any two Lagrangian

potentials for a give neo-classical form Ī must be basic.

Consider one such Lagrangian potential Ī, semibasic over Q. Then at each

n ā—

point m ā M , one may regard Īm as an element of (Tq(m) Q), an n-form at

the corresponding point of Q. This deļ¬nes a map

n

(T ā— Q),

Ī½:M ā’

commuting with the natural projections to Q. Counting dimensions shows that

n

(T ā— Q); to be

if Ī½ is an immersion, then we actually obtain a hypersurface in

more precise, we have a smoothly varying ļ¬eld of hypersurfaces in the vector

bundle n (T ā— Q) ā’ Q. It is not hard to see that Ī½ is an immersion if the

PoincarĀ“-Cartan form Ī is non-degenerate, which is a standing hypothesis. We

e

n

(T ā— Q) over an

can work backwards, as well: given a hypersurface M ā’

(n + 1)-dimensional manifold Q, we may restrict to M the tautological n-form

n

(T ā— Q) to obtain a form Ī ā ā„¦n (M ). Under mild technical hypotheses on

on

the hypersurface M , the form dĪ ā ā„¦n+1(M ) will be a neo-classical PoincarĀ“- e

Cartan form.

So we have associated to a PoincarĀ“-Cartan form Ī , and a choice of La-

e

n

(T ā— Q) ā’ Q. How-

n0

grangian potential Ī ā F ā„¦ , a ļ¬eld of hypersurfaces in

ever, we noted that Ī was not canonically deļ¬ned in terms of Ī , so neither

are these hypersurfaces. As we have seen, the ambiguity in Ī is that another

Ė

admissible Ī may diļ¬er from Ī by a form that is basic over Q. This means

Ė

that Ī ā’ Ī does not depend on the ļ¬ber-coordinate for M ā’ Q, and therefore

4 Thisstatement is only valid in case the foliation by integral manifolds of JĪ is simple; in

other cases, only a cumbersome local version of the statement holds.

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58 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

the two corresponding immersions Ī½, Ī½ diļ¬er in each ļ¬ber Mq (q ā Q) only by

Ė

n

(Tq Q). Consequently, we have in each n (Tq Q) a hyper-

ā— ā—

a translation in

surface well-deļ¬ned up to translation. A contact transformation of M which

preserves Ī will therefore carry the ļ¬eld of hypersurfaces for a particular choice

of Ī to a ļ¬eld of hypersurfaces diļ¬ering by (a ļ¬eld of) aļ¬ne transformations.

To summarize,

one can canonically associate to any neo-classical PoincarĀ“-Cartan

e

n ā—

(T Q) ā’ Q,

form (M, Ī ) a ļ¬eld of hypersurfaces in the bundle

regarded as a bundle of aļ¬ne spaces. We expect the diļ¬erential

invariants of Ī to include information about the geometry of each of

these aļ¬ne hypersurfaces, and this will turn out to be the case.

2.3 Digression on Aļ¬ne Geometry of Hypersur-

faces

Let An+1 denote (n+1)-dimensional aļ¬ne space, which is simply Rn+1 regarded

as a homogeneous space of the group A(n + 1) of aļ¬ne transformations

g ā GL(n + 1, R), v ā Rn+1 .

x ā’ g Ā· x + v,

Let x : F ā’ An+1 denote the principal GL(n + 1, R)-bundle of aļ¬ne frames;

that is,

F = {f = (x, (e0, . . . , en))},

where x ā An+1 is a point, and (e0 , . . . , en) is a basis for the tangent space

Tx An+1. The action is given by

def

a b b

(x, (e0, . . . , en)) Ā· (gb ) = (x, (ebg0 , . . . , eb gn)). (2.24)

For this section, we adopt the index ranges 0 ā¤ a, b, c ā¤ n, 1 ā¤ i, j, k ā¤ n, and

always assume n ā„ 2.

There is a basis of 1-forms Ļ a , Ļ•a on F deļ¬ned by decomposing the An+1 -

b

valued 1-forms

dx = ea Ā· Ļa , dea = eb Ā· Ļ•b .

a

These equations implicitly use a trivialization of T An+1 that commutes with

aļ¬ne transformations. Diļ¬erentiating, we obtain the structure equations for F:

dĻa = ā’Ļ•a ā§ Ļb , dĻ•a = ā’Ļ•a ā§ Ļ•c . (2.25)

b b c b

Choosing a reference frame f0 ā F determines an identiļ¬cation F ā¼ A(n + 1),

=

a a

and under this identiļ¬cation the 1-forms Ļ , Ļ•b on F correspond to a basis

of left-invariant 1-forms on the Lie group A(n + 1). The structure equations

(2.25) on F then correspond to the usual Maurer-Cartan structure equations

for left-invariant 1-forms on a Lie group.

2.3. DIGRESSION ON AFFINE GEOMETRY OF HYPERSURFACES 59

In this section, we will study the geometry of smooth hypersurfaces M n ā‚

An+1 , to be called aļ¬ne hypersurfaces, using the method of moving frames;

no previous knowledge of this method is assumed. In particular, we give con-

structions that associate to M geometric objects in a manner invariant under

aļ¬ne transformations of the ambient An+1 . Among these objects are tensor

ļ¬elds Hij , U ij , and Tijk on M , called the aļ¬ne ļ¬rst and second fundamental

forms and the aļ¬ne cubic form of the hypersurface. We will classify those non-

degenerate (to be deļ¬ned) hypersurfaces for which Tijk = 0 everywhere. This is

of interest because the particular neo-classical PoincarĀ“-Cartan forms that we

e

study later induce ļ¬elds of aļ¬ne hypersurfaces of this type.

Suppose given a smooth aļ¬ne hypersurface M ā‚ An+1 . We deļ¬ne the

collection of 0-adapted frames along M by

F0 (M ) = {(x, (e0, . . . , en )) ā F : x ā M, e1 , . . . , en span Tx M } ā‚ F.

This is a principal subbundle of F|M whose structure group is5

a 0

def

: a ā Rā— , A ā GL(n, R), v ā Rn .

G0 = g0 = (2.26)

v A

Restricting forms on F to F0 (M ) (but supressing notation), we have

Ļ0 = 0, Ļ1 ā§ Ā· Ā· Ā· ā§ Ļn = 0.

Diļ¬erentiating the ļ¬rst of these gives

0 = dĻ0 = ā’Ļ•0 ā§ Ļi ,

i

and we apply the Cartan lemma to obtain

Ļ•0 = Hij Ļj for some functions Hij = Hji.

i

One way to understand the meaning of these functions Hij , which constitute

the ļ¬rst fundamental form of M ā‚ An+1 , is as follows. At any given point of

M ā‚ An+1 , one can ļ¬nd an aļ¬ne frame and associated coordinates with respect

to which M is locally a graph

ĀÆ

x0 = 1 Hij (x1, . . . , xn)xi xj

2

ĀÆ

for some functions Hij . Restricted to the 0-adapted frame ļ¬eld deļ¬ned by

ā‚ ā‚ ā‚

ĀÆ ĀÆ

, ei (x) = (Hij (x)xj + 2 ā‚i Hjk(x)xj xk ) 0 +

1

e0 (x) =

ĀÆ ĀÆ ,

ā‚x0 ā‚xi

ā‚x

ĀÆ

one ļ¬nds that the values over 0 ā M of the functions Hij equal Hij (0). Loosely

speaking, the functions Hij express the second derivatives of a deļ¬ning function

for M .

and throughout, Rā— denotes the connected group of positive real numbers under

5 Here

multiplication.

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60 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

Returning to the general situation, we calculate as follows. We substitute

the expression Ļ•0 = Hij Ļj into the structure equation dĻ•0 = ā’Ļ•0 ā§ Ļ•b , collect

i i i

b

terms, and conclude

0 = (dHij + Hij Ļ•0 ā’ Hkj Ļ•k ā’ HikĻ•k ) ā§ Ļj .

0 i j

Using the Cartan lemma, we have

dHij = ā’Hij Ļ•0 + Hkj Ļ•k + Hik Ļ•k + Tijk Ļk

0 i j

for some functions Tijk = Tikj = Tkji. This inļ¬nitesimally describes how the

functions Hij vary along the ļ¬bers of F0(M ), on which Ļj = 0. In particular,

as a matrix-valued function H = (Hij ) on F0 (M ), it transforms by a linear

ńņš. 14 |