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it says precisely that the complex
d d
F p „¦0 ’’ F p „¦1 ’’ · · ·
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56 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

is locally exact for each p. Now, any almost-classical form Π lies in F n „¦1; so
not only is the closed form Π locally equal to dΛ for some Λ ∈ „¦n (M ), we
can actually choose Λ to lie in F n „¦0. In other words, we can locally ¬nd a
Lagrangian Λ of the form

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

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

Λ = (L0 + pi Li )dx + θ § (Li dx(i)),

and then the condition θ § dΛ = 0 (recall that this was part of the construction
of the Poincar´-Cartan form associated to any class in H n („¦— /I)) gives the
e
relation
‚L
Li (x, z, p) = (x, z, p).
‚pi
This is exactly the condition for Λ to locally be a classical Lagrangian.

Returning to the geometry associated to a neo-classical Poincar´-Cartan e
form Π, we have found (or in case n = 3, postulated) an integrable Pfa¬an sys-
tem JΠ which is invariant under contact transformations preserving Π. Locally
in M , the induced foliation has a smooth “leaf-space” Q of dimension n + 1,
and there is a smooth submersion q : M ’ Q whose ¬bers are n-dimensional
integral manifolds of JΠ . On such a neighborhood, the foliation will be called
simple, and as we are only going to consider the local geometry of Π 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, Π)
look like from the point of view of Qn+1. The ¬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
¬bers of T Q. This is easily seen in coordinates as follows. If, as in the preceding
proof, we integrate JΠ as
JΠ = {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 θ ∈ {dxi} (on M , again),
/
i
we must have dz ’ pidx ∈ “(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 ¬ber coordinates for M ’ Q, and we can map
M ’ Gn(T Q) by

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

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 di¬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 Π is the prolongation of a point transformation of Q, be-
cause the foliation by integral manifolds of JΠ de¬ning Q is associated to Π
in a contact-invariant manner.4 In this sense, studying the geometry of a neo-
classical Poincar´-Cartan form (in case n ≥ 3) under contact transformations is
e
locally no di¬erent than studying the geometry of an equivalence class of classi-
cal non-degenerate ¬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 Π. What kind of geometry does Π de¬ne in terms of Q? We
will answer this question in terms of the following notion.
De¬nition 2.5 A Lagrangian potential for a neo-classical Poincar´-Cartan
e
n0
form Π on M is an n-form Λ ∈ F „¦ (that is, Λ is semibasic for M ’ Q) such
that dΛ = Π.
We saw in the proof of Proposition 2.3 that locally a Lagrangian potential Λ
exists. Such Λ are not unique, but are determined only up to addition of closed
forms in F n „¦0. It will be important below to note that a closed form in F n „¦0
must actually be basic for M ’ Q; that is, it must be locally the pull-back of a
(closed) n-form on Q. In particular, the di¬erence between any two Lagrangian
potentials for a give neo-classical form Π must be basic.
Consider one such Lagrangian potential Λ, semibasic over Q. Then at each
n —
point m ∈ M , one may regard Λm as an element of (Tq(m) Q), an n-form at
the corresponding point of Q. This de¬nes a map
n
(T — Q),
ν:M ’

commuting with the natural projections to Q. Counting dimensions shows that
n
(T — Q); to be
if ν is an immersion, then we actually obtain a hypersurface in
more precise, we have a smoothly varying ¬eld of hypersurfaces in the vector
bundle n (T — Q) ’ Q. It is not hard to see that ν is an immersion if the
Poincar´-Cartan form Π is non-degenerate, which is a standing hypothesis. We
e
n
(T — Q) over an
can work backwards, as well: given a hypersurface M ’
(n + 1)-dimensional manifold Q, we may restrict to M the tautological n-form
n
(T — Q) to obtain a form Λ ∈ „¦n (M ). Under mild technical hypotheses on
on
the hypersurface M , the form dΛ ∈ „¦n+1(M ) will be a neo-classical Poincar´- e
Cartan form.
So we have associated to a Poincar´-Cartan form Π, and a choice of La-
e
n
(T — Q) ’ Q. How-
n0
grangian potential Λ ∈ F „¦ , a ¬eld of hypersurfaces in
ever, we noted that Λ was not canonically de¬ned in terms of Π, so neither
are these hypersurfaces. As we have seen, the ambiguity in Λ is that another
˜
admissible Λ may di¬er from Λ by a form that is basic over Q. This means
˜
that Λ ’ Λ does not depend on the ¬ber-coordinate for M ’ Q, and therefore
4 Thisstatement is only valid in case the foliation by integral manifolds of JΠ 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 ν, ν di¬er in each ¬ber Mq (q ∈ Q) only by
˜
n
(Tq Q). Consequently, we have in each n (Tq Q) a hyper-
— —
a translation in
surface well-de¬ned up to translation. A contact transformation of M which
preserves Π will therefore carry the ¬eld of hypersurfaces for a particular choice
of Λ to a ¬eld of hypersurfaces di¬ering by (a ¬eld of) a¬ne transformations.
To summarize,

one can canonically associate to any neo-classical Poincar´-Cartan
e
n —
(T Q) ’ Q,
form (M, Π) a ¬eld of hypersurfaces in the bundle
regarded as a bundle of a¬ne spaces. We expect the di¬erential
invariants of Π to include information about the geometry of each of
these a¬ne hypersurfaces, and this will turn out to be the case.



2.3 Digression on A¬ne Geometry of Hypersur-
faces
Let An+1 denote (n+1)-dimensional a¬ne space, which is simply Rn+1 regarded
as a homogeneous space of the group A(n + 1) of a¬ne transformations

g ∈ GL(n + 1, R), v ∈ Rn+1 .
x ’ g · x + v,

Let x : F ’ An+1 denote the principal GL(n + 1, R)-bundle of a¬ne frames;
that is,
F = {f = (x, (e0, . . . , en))},
where x ∈ 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)) · (gb ) = (x, (ebg0 , . . . , eb gn)). (2.24)

For this section, we adopt the index ranges 0 ¤ a, b, c ¤ n, 1 ¤ i, j, k ¤ n, and
always assume n ≥ 2.
There is a basis of 1-forms ω a , •a on F de¬ned by decomposing the An+1 -
b
valued 1-forms
dx = ea · ωa , dea = eb · •b .
a

These equations implicitly use a trivialization of T An+1 that commutes with
a¬ne transformations. Di¬erentiating, we obtain the structure equations for F:

dωa = ’•a § ωb , d•a = ’•a § •c . (2.25)
b b c b

Choosing a reference frame f0 ∈ F determines an identi¬cation F ∼ A(n + 1),
=
a a
and under this identi¬cation the 1-forms ω , •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 ‚
An+1 , to be called a¬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
a¬ne transformations of the ambient An+1 . Among these objects are tensor
¬elds Hij , U ij , and Tijk on M , called the a¬ne ¬rst and second fundamental
forms and the a¬ne cubic form of the hypersurface. We will classify those non-
degenerate (to be de¬ned) hypersurfaces for which Tijk = 0 everywhere. This is
of interest because the particular neo-classical Poincar´-Cartan forms that we
e
study later induce ¬elds of a¬ne hypersurfaces of this type.

Suppose given a smooth a¬ne hypersurface M ‚ An+1 . We de¬ne the
collection of 0-adapted frames along M by

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

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

a 0
def
: a ∈ R— , A ∈ GL(n, R), v ∈ Rn .
G0 = g0 = (2.26)
v A

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

ω0 = 0, ω1 § · · · § ωn = 0.

Di¬erentiating the ¬rst of these gives

0 = dω0 = ’•0 § ωi ,
i

and we apply the Cartan lemma to obtain

•0 = Hij ωj for some functions Hij = Hji.
i

One way to understand the meaning of these functions Hij , which constitute
the ¬rst fundamental form of M ‚ An+1 , is as follows. At any given point of
M ‚ An+1 , one can ¬nd an a¬ne frame and associated coordinates with respect
to which M is locally a graph
¯
x0 = 1 Hij (x1, . . . , xn)xi xj
2

¯
for some functions Hij . Restricted to the 0-adapted frame ¬eld de¬ned by
‚ ‚ ‚
¯ ¯
, ei (x) = (Hij (x)xj + 2 ‚i Hjk(x)xj xk ) 0 +
1
e0 (x) =
¯ ¯ ,
‚x0 ‚xi
‚x
¯
one ¬nds that the values over 0 ∈ M of the functions Hij equal Hij (0). Loosely
speaking, the functions Hij express the second derivatives of a de¬ning function
for M .
and throughout, R— 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 •0 = Hij ωj into the structure equation d•0 = ’•0 § •b , collect
i i i
b
terms, and conclude
0 = (dHij + Hij •0 ’ Hkj •k ’ Hik•k ) § ωj .
0 i j

Using the Cartan lemma, we have
dHij = ’Hij •0 + Hkj •k + Hik •k + Tijk ωk
0 i j

for some functions Tijk = Tikj = Tkji. This in¬nitesimally describes how the
functions Hij vary along the ¬bers of F0(M ), on which ωj = 0. In particular,
as a matrix-valued function H = (Hij ) on F0 (M ), it transforms by a linear

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( 48 .)



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