<< Предыдущая стр. 14(из 48 стр.)ОГЛАВЛЕНИЕ Следующая >>
it says precisely that the complex
d d
F p в„¦0 в€’в†’ F p в„¦1 в€’в†’ В· В· В·
Вґ
56 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

is locally exact for each p. Now, any almost-classical form О  lies in F n в„¦1; so
not only is the closed form О  locally equal to dО› for some О› в€€ в„¦n (M ), we
can actually choose О› to lie in F n в„¦0. In other words, we can locally п¬Ѓnd a
Lagrangian О› of the form

О› = L0 (x, z, p)dx + Li (x, z, p)dz в€§ dx(i)

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

О› = (L0 + pi Li )dx + Оё в€§ (Li dx(i)),

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

Returning to the geometry associated to a neo-classical PoincarВґ-Cartan e
form О , we have found (or in case n = 3, postulated) an integrable Pfaп¬ѓan sys-
tem JО  which is invariant under contact transformations preserving О . Locally
in M , the induced foliation has a smooth вЂњleaf-spaceвЂќ Q of dimension n + 1,
and there is a smooth submersion q : M в†’ Q whose п¬Ѓbers are n-dimensional
integral manifolds of JО  . On such a neighborhood, the foliation will be called
simple, and as we are only going to consider the local geometry of О  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, О )
look like from the point of view of Qn+1. The п¬Ѓrst 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
п¬Ѓbers of T Q. This is easily seen in coordinates as follows. If, as in the preceding
proof, we integrate JО  as
JО  = {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 Оё в€€ {dxi} (on M , again),
/
i
we must have dz в€’ pidx в€€ О“(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 п¬Ѓber coordinates for M в†’ Q, and we can map
M в†’ Gn(T Q) by

(xi, z, pi) в†’ ((xi , z); {dz в€’ pi dxi}вЉҐ ).

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 diп¬Ђeomorphism commuting with projections to
Вґ
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 О  is the prolongation of a point transformation of Q, be-
cause the foliation by integral manifolds of JО  deп¬Ѓning Q is associated to О
in a contact-invariant manner.4 In this sense, studying the geometry of a neo-
classical PoincarВґ-Cartan form (in case n в‰Ґ 3) under contact transformations is
e
locally no diп¬Ђerent than studying the geometry of an equivalence class of classi-
cal non-degenerate п¬Ѓrst-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 О . What kind of geometry does О  deп¬Ѓne in terms of Q? We
will answer this question in terms of the following notion.
Deп¬Ѓnition 2.5 A Lagrangian potential for a neo-classical PoincarВґ-Cartan
e
n0
form О  on M is an n-form О› в€€ F в„¦ (that is, О› is semibasic for M в†’ Q) such
that dО› = О .
We saw in the proof of Proposition 2.3 that locally a Lagrangian potential О›
exists. Such О› are not unique, but are determined only up to addition of closed
forms in F n в„¦0. It will be important below to note that a closed form in F n в„¦0
must actually be basic for M в†’ Q; that is, it must be locally the pull-back of a
(closed) n-form on Q. In particular, the diп¬Ђerence between any two Lagrangian
potentials for a give neo-classical form О  must be basic.
Consider one such Lagrangian potential О›, semibasic over Q. Then at each
n в€—
point m в€€ M , one may regard О›m as an element of (Tq(m) Q), an n-form at
the corresponding point of Q. This deп¬Ѓnes a map
n
(T в€— Q),
ОЅ:M в†’

commuting with the natural projections to Q. Counting dimensions shows that
n
(T в€— Q); to be
if ОЅ is an immersion, then we actually obtain a hypersurface in
more precise, we have a smoothly varying п¬Ѓeld of hypersurfaces in the vector
bundle n (T в€— Q) в†’ Q. It is not hard to see that ОЅ is an immersion if the
PoincarВґ-Cartan form О  is non-degenerate, which is a standing hypothesis. We
e
n
(T в€— Q) over an
can work backwards, as well: given a hypersurface M в†’
(n + 1)-dimensional manifold Q, we may restrict to M the tautological n-form
n
(T в€— Q) to obtain a form О› в€€ в„¦n (M ). Under mild technical hypotheses on
on
the hypersurface M , the form dО› в€€ в„¦n+1(M ) will be a neo-classical PoincarВґ- e
Cartan form.
So we have associated to a PoincarВґ-Cartan form О , and a choice of La-
e
n
(T в€— Q) в†’ Q. How-
n0
grangian potential О› в€€ F в„¦ , a п¬Ѓeld of hypersurfaces in
ever, we noted that О› was not canonically deп¬Ѓned in terms of О , so neither
are these hypersurfaces. As we have seen, the ambiguity in О› is that another
Лњ
admissible О› may diп¬Ђer from О› by a form that is basic over Q. This means
Лњ
that О› в€’ О› does not depend on the п¬Ѓber-coordinate for M в†’ Q, and therefore
4 Thisstatement is only valid in case the foliation by integral manifolds of JО  is simple; in
other cases, only a cumbersome local version of the statement holds.
Вґ
58 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

the two corresponding immersions ОЅ, ОЅ diп¬Ђer in each п¬Ѓber Mq (q в€€ Q) only by
Лњ
n
(Tq Q). Consequently, we have in each n (Tq Q) a hyper-
в€— в€—
a translation in
surface well-deп¬Ѓned up to translation. A contact transformation of M which
preserves О  will therefore carry the п¬Ѓeld of hypersurfaces for a particular choice
of О› to a п¬Ѓeld of hypersurfaces diп¬Ђering by (a п¬Ѓeld of) aп¬ѓne transformations.
To summarize,

one can canonically associate to any neo-classical PoincarВґ-Cartan
e
n в€—
(T Q) в†’ Q,
form (M, О ) a п¬Ѓeld of hypersurfaces in the bundle
regarded as a bundle of aп¬ѓne spaces. We expect the diп¬Ђerential
invariants of О  to include information about the geometry of each of
these aп¬ѓne hypersurfaces, and this will turn out to be the case.

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

g в€€ GL(n + 1, R), v в€€ Rn+1 .
x в†’ g В· x + v,

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

For this section, we adopt the index ranges 0 в‰¤ a, b, c в‰¤ n, 1 в‰¤ i, j, k в‰¤ n, and
always assume n в‰Ґ 2.
There is a basis of 1-forms П‰ a , П•a on F deп¬Ѓned by decomposing the An+1 -
b
valued 1-forms
dx = ea В· П‰a , dea = eb В· П•b .
a

These equations implicitly use a trivialization of T An+1 that commutes with
aп¬ѓne transformations. Diп¬Ђerentiating, we obtain the structure equations for F:

dП‰a = в€’П•a в€§ П‰b , dП•a = в€’П•a в€§ П•c . (2.25)
b b c b

Choosing a reference frame f0 в€€ F determines an identiп¬Ѓcation F в€ј A(n + 1),
=
a a
and under this identiп¬Ѓcation the 1-forms П‰ , П•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 вЉ‚
An+1 , to be called aп¬ѓne 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
aп¬ѓne transformations of the ambient An+1 . Among these objects are tensor
п¬Ѓelds Hij , U ij , and Tijk on M , called the aп¬ѓne п¬Ѓrst and second fundamental
forms and the aп¬ѓne cubic form of the hypersurface. We will classify those non-
degenerate (to be deп¬Ѓned) hypersurfaces for which Tijk = 0 everywhere. This is
of interest because the particular neo-classical PoincarВґ-Cartan forms that we
e
study later induce п¬Ѓelds of aп¬ѓne hypersurfaces of this type.

Suppose given a smooth aп¬ѓne hypersurface M вЉ‚ An+1 . We deп¬Ѓne the
collection of 0-adapted frames along M by

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

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

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

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

П‰0 = 0, П‰1 в€§ В· В· В· в€§ П‰n = 0.

Diп¬Ђerentiating the п¬Ѓrst of these gives

0 = dП‰0 = в€’П•0 в€§ П‰i ,
i

and we apply the Cartan lemma to obtain

П•0 = Hij П‰j for some functions Hij = Hji.
i

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

ВЇ
for some functions Hij . Restricted to the 0-adapted frame п¬Ѓeld deп¬Ѓned by
в€‚ в€‚ в€‚
ВЇ ВЇ
, ei (x) = (Hij (x)xj + 2 в€‚i Hjk(x)xj xk ) 0 +
1
e0 (x) =
ВЇ ВЇ ,
в€‚x0 в€‚xi
в€‚x
ВЇ
one п¬Ѓnds that the values over 0 в€€ M of the functions Hij equal Hij (0). Loosely
speaking, the functions Hij express the second derivatives of a deп¬Ѓning function
for M .
and throughout, Rв€— denotes the connected group of positive real numbers under
5 Here

multiplication.
Вґ
60 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

Returning to the general situation, we calculate as follows. We substitute
the expression П•0 = Hij П‰j into the structure equation dП•0 = в€’П•0 в€§ П•b , collect
i i i
b
terms, and conclude
0 = (dHij + Hij П•0 в€’ Hkj П•k в€’ HikП•k ) в€§ П‰j .
0 i j

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

for some functions Tijk = Tikj = Tkji. This inп¬Ѓnitesimally describes how the
functions Hij vary along the п¬Ѓbers of F0(M ), on which П‰j = 0. In particular,
as a matrix-valued function H = (Hij ) on F0 (M ), it transforms by a linear
 << Предыдущая стр. 14(из 48 стр.)ОГЛАВЛЕНИЕ Следующая >>