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it says precisely that the complex
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 āleaf-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 Īm 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· Ļa , 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:

dĻa = ā’Ļ•a ā§ Ļb , 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 ā¼ 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

Ļ0 = 0, Ļ1 ā§ Ā· Ā· Ā· ā§ Ļn = 0.

Diļ¬erentiating the ļ¬rst of these gives

0 = dĻ0 = ā’Ļ•0 ā§ Ļi ,
i

and we apply the Cartan lemma to obtain

Ļ•0 = Hij Ļj 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 Ļj 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§ Ļj .
0 i j

Using the Cartan lemma, we have
dHij = ā’Hij Ļ•0 + Hkj Ļ•k + Hik Ļ•k + Tijk Ļk
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 Ļj = 0. In particular,
as a matrix-valued function H = (Hij ) on F0 (M ), it transforms by a linear
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