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• dU ij ≡ 2nρU ij ’ ±i U lj ’ ±j U il ,
l l

• dSj ≡ (n ’ 2)ρSj ’ ±iSj + Sli ±l + 1 (U il σlj + U jl σli ) ’ n δj U kl σkl + T ijk δk ,
1i
i i l
j
l 2

all modulo {θ, ωi , πi}. Notice in particular that T ijk and U ij transform by a
combination of rescaling and a standard representation of SO(n). However,
(Sj ) is only a tensor when the tensors (T ijk ) and (U ij ) both vanish. We will
i

consider this situation in the next chapter.
An interpretation of the ¬rst two transformation rules is that the objects
2
T ijk (πi —¦ πj —¦ πk ) — |π1 § · · · § πn |’ n ,
T =
U ij πi —¦ πj
U=

are invariant modulo JΠ = {θ, ωi } under ¬‚ows along ¬bers over M ; that is, when
restricted to a ¬ber of B2 ’ Q, they actually descend to well-de¬ned objects on
the smaller ¬ber of M ’ Q. The restriction to ¬bers suggests our next result,
which nicely relates the di¬erential invariants of the Poincar´-Cartan form with
e
the a¬ne geometry of hypersurfaces discussed in the preceding section.

Theorem 2.4 The functions T ijk and U ij are coe¬cients of the a¬ne cubic
form and a¬ne second fundamental form for the ¬berwise a¬ne hypersurfaces
n
(T — Q) induced by a semibasic Lagrangian potential Λ of Π.
in

Proving this is a matter of identifying the bundles where the two sets of invari-
ants are de¬ned, and unwinding the de¬nitions.
In the next section, we will brie¬‚y build on the preceding results in the case
where T ijk = 0 and U ij = 0, showing that these conditions roughly characterize
those de¬nite neo-classical Poincar´-Cartan forms appearing in the problem of
e
¬nding prescribed mean curvature hypersurfaces, in Riemannian or Lorentzian
manifolds. In the next chapter, we will extensively consider the case T ijk = 0,
U ij = 0, which includes remarkable Poincar´-Cartan forms arising in conformal
e
ijk
geometry. About the case for which T = 0, nothing is known.
For reference, we summarize the results of the equivalence method that will
be used below. Associated to a de¬nite, neo-classical Poincar´-Cartan form Π
e
´
74 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

on a contact manifold (M, I) is a G-structure B ’ M , where
±« 

n’2
 ±r 0 0 
(Ai ) ∈ SO(n, R), r > 0,
’2 i
G=  : j
0 r Aj 0 .
Sij = Sji, Sii = 0
 
’1 j
Sik Ak n
±r (A )i
Di j
(2.47)

B ’ M supports a pseudo-connection (not uniquely determined)
« 
(n ’ 2)ρ 0 0
• = ’ ,
’2ρδj + ±i
i
0 0
j
j j
nρδi ’ ±i
δi σij

with ±i + ±j = 0, σij = σji, σii = 0, such that in the structure equation
j i
«  « 
θ θ
d  ωi  = ’• §  ωj  + „,
πi πj

the torsion is of the form
« 
’πi § ωi
„ =  ’(Sj ωj + U ij πj ) § θ + T ijk πj § ωk  ,
i

0

with
j
T ijk = T jik = T kji, T iik = 0; U ij = U ji ; Sj = Si , Si = 0.
i i


In terms of any section of B ’ M , the Poincar´-Cartan form is
e

Π = ’θ § πi § ω(i).

One further structure equation is

(n ’ 2)dρ = ’δi § ωi ’ Sj πi § ωj +
i
U ij σij § θ ’ ti πi § θ.
n’2
(2.48)
2n




2.5 The Prescribed Mean Curvature System
In this section, we will give an application of the part of the equivalence method
completed so far. We will show that a de¬nite, neo-classical Poincar´-Cartan
e
ijk
form with T = 0, and satisfying an additional open condition speci¬ed below,
is locally equivalent to that which arises in the problem of ¬nding in a given
Riemannian manifold a hypersurface whose mean curvature coincides with a
prescribed background function. This conclusion is presented as Theorem 2.5.
2.5. THE PRESCRIBED MEAN CURVATURE SYSTEM 75

To obtain this result, we continue applying the equivalence method where
we left o¬ in the preceding section, and take up the case T ijk = 0. From our
calculations in a¬ne hypersurface geometry, we know that this implies that
U ij = »δ ij ,
for some function » on the principal bundle B ’ M ; alternatively, this can
be shown by computations continuing those of the preceding section. We will
show that under the hypothesis » < 0, the Poincar´-Cartan form Π is locally
e
equivalent to that occuring in a prescribed mean curvature system.
We have in general on B that
dU ij ≡ 2nρU ij ’ ±i U kj ’ ±j U ik (mod {θ, ωi , πi}).
k k

Then for our U ij = »δ ij , the function » scales positively along ¬bers of B ’ M ,
so under our assumption » < 0 we may make a reduction to
B1 = {u ∈ B : »(u) = ’1} ‚ B;
this de¬nes a subbundle of B of codimension 1, on which ρ is semibasic over
M .7 In particular, on B1 we may write
H
θ + E i ω i + F i πi
ρ=’
2n
for some functions H, Ei , F i. The reason for the normalization of the θ-
coe¬cient will appear shortly.
We claim that F i = 0. To see this, start from the equation (2.48) for dρ,
which on B1 reads
(n ’ 2)dρ = ’δi § ωi ’ ti πi § θ ’ Sj πi § ωj .
i


Then, as we have done so often, we compute d2ω, where ω = ω1 § · · · § ωn and
dωi = 2ρ § ωi ’ ±i § ωj + πi § θ ’ Sj ωj § θ.
i
j

We ¬nd
dω = 2nρ § ω ’ θ § πi § ω(i) = 2nρ § ω + Π,
and the next step is simpli¬ed by knowing dΠ = 0:
d2 ω
0 =
2n dρ § ω ’ 2nρ § dω
=
(ti πi § θ) § ω
2n

= n’2

+2n(Ej ωj + F j πj ) § θ § πi § ω(i)
ti
2nθ § F j πi § πj § ω(i) + + E i πi § ω .
=
n’2
7 In
this section, we will denote by B1 , B2 , etc., successive reductions of the G-structure
B ’ M which was constructed in the preceding section. These are not the same as the
bundles of the same name used in constructing B, which are no longer needed.
´
76 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

This gives our claim F i = 0, as well as

(n ’ 2)E i = ’ti .

For our next reduction, we will show that we can de¬ne a principal subbundle

B2 = {u ∈ B1 : E i (u) = 0} ‚ B1 ,

having structure group de¬ned by the condition Di = 0, r = 1 in (2.47). This
2
{θ, ωi , πi}:
follows by computing modulo

(n ’ 2)dρ ≡ ’δi § ωi ,

and also
dρ ≡ ’ 2n dH § θ + dEi § ωi ’ Ej ±j § ωi .
1
i

Comparing these, we obtain

dEi ’ Ej ±j + 1
§ ωi ’ 1
§ θ ≡ 0.
n’2 δi 2n dH
i


This implies that

dEi ’ Ej ±j + ≡ 0 (mod {θ, ωi , πi}),
1
n’2 δi
i

justifying the described reduction to B2 ’ M , on which ρ and δi are semibasic.
Finally, a third reduction is made possible by the general equation

dSj ≡ (n ’ 2)ρSj ’ ±i Sj + Sk ±k + 1 (U il σlj + U jl σli ) ’ n δj U kl σkl,
i i k i 1i
k j 2

modulo {θ, ωi , πi}. On B2 , where in particular » = ’1 and ρ is semibasic, we
have
dSj ≡ ’±i Sj + Sk ±k ’ σij (mod {θ, ωi , πi}).
i k i
k j
i
This means that the torsion matrix (Sj ) can undergo translation by an arbitrary
traceless symmetric matrix along the ¬bers of B2 ’ M , so the locus
i
B3 = {u ∈ B2 : Sj (u) = 0} ‚ B2

is a subbundle, whose structure group is SO(n, R) with Lie algebra represented
by matrices of the form
« 
00 0
±i + ±j = 0.
a2 =  0 ± i 0 ,
j j i
0 0 ’±j i

This is all the reduction that we shall need. On B3 , we have equations
± H
 ρ = ’ 2n θ,
dθ = ’πi § ωi (because ρ § θ = 0 on B3 ),

(n ’ 2)dρ = ’δi § ωi (because ti = ’(n ’ 2)E i = 0 on B3 ).
2.5. THE PRESCRIBED MEAN CURVATURE SYSTEM 77

The δi appearing the third equation are semibasic over M , and the three equa-
tions together imply that
dH ≡ 0 (mod {θ, ω i }).
This last observation is quite important. Recall the integrable Pfa¬an system
JΠ = {θ, ωi }, assumed to have a well-de¬ned leaf-space Qn+1 with submersion
M ’ Q. The last equation shows that H is locally constant along the ¬bers of
M ’ Q, and may therefore be thought of as a function on Q.
Now, considering the two structure equations

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. 18
( 48 .)



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