• 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