Again working modulo Rk’1, the only term that is non-linear in (pK , pI ) is

‚hK K

so as before, we can assume ‚hA = 0 and write

‚pA pAi pKi , ‚p

g(xi , p(k)) = h(x, p(k’2)) + hK (x, p(k’3))pK + hI (x, p(k’3))pI .

The ¬nal step is similar, and gives that h is linear in pA . This yields

g(xi , p(k)) = h + hApA + hK pK + hI pI , (4.48)

182 CHAPTER 4. ADDITIONAL TOPICS

where each of h, hA, hK , hI is a function of (xi , p(k’3)). This is the ¬rst state-

ment of the proposition.

To derive the second statement, we use the form (4.48) in the condition

(4.46) modulo Rk’2, in which the only term non-linear in (pK , pI ) is

d|I|f

I i (k’3)

0≡ h (x , p )I

dx

hI (xi , p(k’3))f (p)

≡ pI\i pi.

i∈I

In particular, if f (p) = 0, then we must have hI = 0. Therefore g ∈ Rk

actually lies in Rk’1, and we can induct downward on k, eventually proving

that g ∈ R1 , as desired.

We mention two situations which contrast sharply with that of the non-linear

Poisson equation in dimension n ≥ 3. First, in the case of a linear Poisson

equation ∆z = f(z), f (z) = 0 (still in n ≥ 3 dimensions), one can extend the

preceding argument to show that a generating function for a conservation law

is linear in all of the derivative variables pI . In particular, the in¬nite collection

of conservation laws for the Laplace equation ∆z = 0 can be determined in this

manner; it is interesting to see how all of these disappear upon the addition of

a non-linear term to the equation.

Second, in dimension n = 2, there are well-known non-linear Poisson equa-

tions ∆u = sinh u and ∆u = eu having in¬nitely many higher-order conservation

laws, but we will not discuss these.

The K = ’1 Surface System

4.3.3

In §1.4, we constructed Monge-Ampere systems on the contact manifold M 5

of oriented tangent planes to Euclidean space E3 , whose integral manifolds

corresponded to linear Weingarten surfaces. We brie¬‚y recall this setup for

the case of surfaces with Gauss curvature K = ’1. Our index ranges are now

1 ¤ a, b, c ¤ 3, 1 ¤ i, j, k ¤ 2.

Let F ’ E3 be the Euclidean frame bundle, with its global coframing ω a ,

a b

ωb = ’ωa satisfying the structure equations

dωa = ’ωb § ωb ,

a a a c

dωb = ’ωc § ωb .

We set θ = ω3 , πi = ωi , and then θ ∈ „¦1(F) is the pullback of a global contact

3

form on M = G2 (T E3 ). The forms that are semibasic over M are generated by

θ, ωi , πi ∈ „¦1 (F).

We de¬ne the 2-forms on F

˜ = dθ = ’π1 § ω1 ’ π2 § ω2 ,

Ψ = π 1 § π2 + ω 1 § ω 2 ,

which are pullbacks of uniquely determined forms on M . On a transverse inte-

gral element E 2 ‚ Tm M of the contact system I = {θ, ˜}, on which ω 1 §ω2 = 0,

4.3. HIGHER-ORDER CONSERVATION LAWS 183

there are relations

πi = hij ωj , hij = hji.

In this case,

π1 § π2 = Kω1 § ω2 ,

where K = h11h22 ’ h12h21 is the Gauss curvature of any surface N 2 ’ E3

whose 1-jet graph in M 5 is tangent to E ‚ Tm M . Therefore, transverse integral

manifolds of the EDS

E = {θ, ˜, Ψ} (4.49)

correspond locally to surfaces in E3 with constant Gauss curvature K = ’1.

The EDS (M, E) is an example of a hyperbolic Monge-Ampere system; this

notion appeared in §2.1, where we used it to specify a branch of the equiva-

lence problem for Poincar´-Cartan forms on contact 5-manifolds. The de¬ning

e

property of a hyperbolic Monge-Ampere system E = {θ, ˜, Ψ} is that modulo

the algebraic ideal {θ}, E contains two distinct (modulo scaling) decomposable

2-forms; that is, one can ¬nd two non-trivial linear combinations of the form

± 1 § β1 ,

»1 ˜ + µ 1 Ψ =

± 2 § β2 .

»2 ˜ + µ 2 Ψ =

This exhibits two rank-2 Pfa¬an systems Ii = {±i, βi }, called the characteristic

systems of E, which are easily seen to be independent of choices (except for which

one is I1 and which one is I2 ). The relationship between the geometry of the

characteristic systems and that of the original hyperbolic Monge-Ampere system

is very rich (see [BGH95]). Of particular interest are those hyperbolic systems

whose characteristic systems each contain a non-trivial conservation law. We

will show that this holds for the K = ’1 system introduced above, but only

after one prolongation. In other words, for the prolonged system E (1) , there is

(1)

also a notion of characteristic systems Ii which restrict to any integral surface

(1)

as the original Ii , and each of these Ii contains a non-trivial conservation law

for E (1).

Returning to the discussion of integral elements of E = {θ, ˜, Ψ}, note that

for any integral element E ‚ T(p,H) M , given by equations

πi ’ hij ωj = 0,

there is a unique frame (p, (e1 , e2, e1 — e2 )) ∈ F over (p, e1 § e2 ) ∈ M for which

the second fundamental form is normalized as

1

h22 = ’ a ,

h11 = a > 0, h12 = h21 = 0.

The tangent lines in E3 spanned by these e1 , e2 are the principal directions at

p of any surface whose 1-jet graph is tangent to E; they de¬ne an orthonormal

frame in which the second fundamental form is diagonal. The number a > 0 is

184 CHAPTER 4. ADDITIONAL TOPICS

determined by the plane E ‚ T M , so to study integral elements of (M, E), and

in particular to calculate on its ¬rst prolongation, we introduce

F (1) = F — R—,

where R— has the coordinate a > 0. There is a projection F (1) ’ M (1) , mapping

(p, e, a) ’ (p, e1 § e2 , {π1 ’ aω1 , π2 + a ω2 }⊥ ).

1

We de¬ne on F (1) the forms

θ1 = π1 ’ aω1 ,

θ2 = π 2 + a ω 2 ,

1

which are semibasic for F (1) ’ M (1). The ¬rst prolongation of the system E

on M is a Pfa¬an system on M (1) , which pulls back to F (1) as

E (1) = {θ, θ1 , θ2, dθ1, dθ2 }.

We have structure equations

dθ ≡ ’θ1 § ω1 ’ θ2 § ω2 ≡ 0

2

dθ1 ≡ ’da § ω1 + 1+a ω2 § ω2

1

(mod {θ, θ1 , θ2}),

a2

dθ2 ≡ a2 da § ω2 + 1+a ω2 § ω1

1

1

a

and in particular, we have the decomposable linear combinations

’dθ1 ’ a dθ2 = (da ’ (1 + a2 )ω2 ) § (ω1 + a ω2 ),

1

1

(4.50)

’dθ1 + a dθ2 = (da + (1 + a2 )ω2 ) § (ω1 ’ a ω2 ).

1

1

The EDS E (1) is algebraically generated by θ, θ1 , θ2 , and these two decomposable

2-forms. The characteristic systems are by de¬nition di¬erentially generated by

(1)

I1 = {θ, θ1 , θ2 , da ’ (1 + a2 )ω2 , ω1 + a ω2 },

1

1

(4.51)

(1)

I2 = {θ, θ1 , θ2 , da + (1 + a2 )ω2 , ω1 ’ a ω2 }.

1

1

Now, the “universal” second fundamental form can be factored as

II = a(ω1 )2 ’ a (ω2 )2 = a(ω1 + a ω2 )(ω1 ’ 1 ω2 ).

1 1

(4.52)

a

These linear factors, restricted any K = ’1 surface, de¬ne the asymptotic curves

of that surface, so by comparing (4.51) and (4.52) we ¬nd that:

On a K = ’1 surface, the integral curves of the characteristic sys-

tems are the asymptotic curves.

(1)

Now we look for Euclidean-invariant conservation laws in each Ii . Instead

of using Noether™s theorem, we work directly. We start by setting

(1)

•1 = f(a)(ω1 + a ω2 ) ∈ I1 ,

1

4.3. HIGHER-ORDER CONSERVATION LAWS 185

and seek conditions on f(a) to have d•1 ∈ E (1). A short computation using the

structure equations gives

d•1 ≡ (f (a)(1 + a2) ’ f(a)a)ω2 § (ω1 + a ω2 )

1

(mod E (1)),

1

so the condition for •1 to be a conserved 1-form is

f (a) a

= .

1 + a2

f(a)

A solution is

1 + a2 (ω1 + 1 ω2);

1

•1 = 2 a

1

the choice of multiplicative constant 2 will simplify later computations. A

similar computation, seeking an appropriate multiple of ω 1 ’ a ω2 , yields the

1

conserved 1-form

•2 = 1 1 + a2 (ω1 ’ 1 ω2).

2 a

On any simply connected integral surface of the K = ’1 system, there are

coordinate functions s, t such that

•1 = ds, •2 = dt.

If we work in these coordinates, and in particular use the non-orthonormal

coframing (•1 , •2) then we can write

1

ω1 √

= (•1 + •2 ), (4.53)

1 + a2

a

ω2 √ (•1 ’ •2 ),

= (4.54)

1 + a2

1 ’ a2

12 22 2

•1 •2 + (•2 )2 ,

I = (ω ) + (ω ) = (•1 ) + 2 (4.55)

1 + a2

4a

II = •1 •2 . (4.56)

1 + a2

These expressions suggest that we de¬ne

a = tan z,

where a > 0 means that we can smoothly choose z = tan’1 a ∈ (0, π/2). Note

that 2z is the angle measure between the asymptotic directions •⊥ , •⊥ , and

1 2

that

ω1 = (cos z)(•1 + •2 ), ω2 = (sin z)(•1 ’ •2 ). (4.57)

The following is fundamental in the study of K = ’1 surfaces.

186 CHAPTER 4. ADDITIONAL TOPICS