K = ’1, the associated function z, expressed in terms of asymptotic coordinates

s, t, satis¬es the sine-Gordon equation

1

zst = sin(2z). (4.58)

2

One can prove this by a direct computation, but we will instead highlight certain

general EDS constructions which relate the K = ’1 di¬erential system to a

hyperbolic Monge-Ampere system associated to the sine-Gordon equation. One

of these is the notion of an integrable extension of an exterior di¬erential system,

which we have not yet encountered. This is a device that handles a forseeable

di¬culty; namely, the sine-Gordon equation is expressed in terms of the variables

s and t, but for the K = ’1 system, these are primitives of a conservation law,

de¬ned on integral manifolds of the system only up to addition of integration

constants. One can think of an integrable extension as a device for appending

the primitives of conserved 1-forms. More precisely, an integrable extention of

π

an EDS (M, E) is given by a submersion M ’ M , with a di¬erential ideal E

on M generated algebraically by π — E and some 1-forms on M . In this case, the

preimage in M of an integral manifold of E is foliated by integral manifolds of

E . For example, if a 1-form • ∈ „¦1 (M ) is a conservation law for E, then one

can take M = M — R, and let E ‚ „¦— (M ) be generated by E and • = • ’ ds,

˜

where s is a ¬ber coordinate on R. Then the preimage in M of any integral

manifold of (M, E) is foliated by a 1-parameter family of integral manifolds of

(M , E ), where the parameter corresponds to a choice of integration constant

for •.10

Proof. Because z is de¬ned only on M (1), it is clear that we will have to prolong

once more to study zst . (Twice is unnecessary, because of the Monge-Ampere

form of (4.58).) From (4.50), we see that integral elements with •1 § •2 = 0

satisfy

1 1

dz ’ ω2 = 2p•1, dz + ω2 = 2q•2,

for some p, q. These p, q can be taken as ¬ber coordinates on the second pro-

longation

M (2) = M (1) — R2.

Let

θ3 = dz ’ p•1 ’ q•2 ,

1

θ4 = ω2 + p•1 ’ q•2,

and then the prolonged di¬erential system is

E (2) = {θ, θ1 , . . . , θ4 , dθ3, dθ4}.

Integral manifolds for the original K = ’1 system correspond to integral man-

ifolds of E (2); in particular, on such an integral manifold f (2) : N ’ M (2) we

have

0 = dθ3|N (2) = ’dp § •1 ’ dq § •2

10 For more information about integrable extensions, see §6 of [BG95b].

4.3. HIGHER-ORDER CONSERVATION LAWS 187

and

dθ4|N (2)

0=

Kω1 § ω2 + dp § •1 ’ dq § •2

=

(’1)(cos z)(•1 + •2) § (sin z)(•1 ’ •2) + dp § •1 ’ dq § •2

=

sin(2z)•1 § •2 + dp § •1 ’ dq § •2 .

=

Now we de¬ne the integrable extension

M (2) = M (2) — R2,

where R2 has coordinates s, t, and on M (2) we de¬ne the EDS E (2) to be

generated by E (2), along with the 1-forms •1 ’ ds, •2 ’ dt. An integral manifold

(2)

f (2) : N ’ M (2) of E (2) gives a 2-parameter family of integral manifolds fs0 ,t0 :

N ’ M (2) of E (2) . On any of these, the functions s, t will be local coordinates,

and we will have

0 = dz ’ p ds ’ q dt,

0 = ’dp § ds ’ dq § dt,

0 = sin(2z)ds § dt ’ ds § dp ’ dq § dt.

These three clearly imply that z(s, t) satis¬es (4.58).

Note that we can start from the other side, de¬ning the di¬erential system

for the sine-Gordon equation as

EsG = {dz ’ p ds ’ q dt, ’dp § ds ’ dq § dt, ds § dp + dq § dt ’ sin(2z)ds § dt},

which is a Monge-Ampere system on the contact manifold J 1(R2 , R). One can

form a “non-abelian” integrable extension

P = J 1(R2 , R) — F

of (J 1(R2 , R), EsG) by taking

±1

ω ’ (cos z)(ds + dt),

2

ω ’ (sin z)(ds + dt),

3

ω,

EP = EsG + .

1

ω2 + p ds ’ q dt,

3

ω ’ (sin z)(ds + dt),

1

3

ω2 + (cos z)(ds ’ dt).

This system is di¬erentially closed, as one can see by using the structure equa-

tions for F and assuming that z satis¬es the sine-Gordon equation. Though the

188 CHAPTER 4. ADDITIONAL TOPICS

following diagram is complicated, it sums up the whole story:

(M (2) , E (2) ) / (P, EP )

OOO

oo OOO

ooo OOO

oo

ooo OOO

wo '

1 2

(2) (2)

(M (1) , E (1) ) (J (R , R), EsG)

(M , E )

OOO

ooo

OOO

ooo

OOO

ooo

OO' wooo

(M (1) , E (1))

(M, E)

The system (M (1) , E (1) ) was not introduced in the proof; it is the integrable

extension of (M (1) , E (1)) formed by adjoining primitives s, t for the conserved 1-

forms •1, •2 , and its prolongation turns out to be (M (2) , E (2) ). In other words,

starting on M (1) , one can ¬rst prolong and then adjoin primitives, or vice versa.

The main point of this diagram is:

The identi¬cation (M (2) , E (2) ) ’’ (P, EP ) is an isomorphism of

exterior di¬erential systems. In other words, modulo prolongations

and integrable extensions, the K = ’1 system and the sine-Gordon

system are equivalent.

Note that while conservation laws are preserved under prolongation, there is

an additional subtlety for integrable extensions. In this case, only those conser-

vation laws for the sine-Gordon system that are invariant under s, t-translation

give conservation laws for the K = ’1 system. Conversely, only those conser-

vation laws for the K = ’1 system that are invariant under Euclidean motions

(i.e., translations in F) give conservation laws for the sine-Gordon system. There

is even a di¬culty involving trivial conservation laws; namely, the non-trivial,

Euclidean-invariant conservation laws •1, •2 for the K = ’1 system induce the

trivial conservation laws ds, dt for the sine-Gordon system.

However, we do have two conservation laws for sine-Gordon, obtained via

Noether™s theorem applied to s, t-translations and the Lagrangian

1

ΛsG = (pq ’ 2 (cos(2z) ’ 1))ds § dt;

they are

ψ1 = 1 p2ds ’ 4 (cos(2z) ’ 1)dt,

1

2

ψ2 = 2 q2dt + 1 (cos(2z) ’ 1)ds.

1

4

The corresponding conserved 1-forms on (M (2), E (2)) are

•3 = 1 p2•1 ’ 4 (cos(2z) ’ 1)•2,

1

2

•4 = 2 q2•2 + 1 (cos(2z) ’ 1)•1 .

1

4

4.3. HIGHER-ORDER CONSERVATION LAWS 189

In the next section, we will introduce a B¨cklund transformation for the sine-

a

Gordon equation, which can be used to generate an in¬nite double-sequence

ψ2k’1, ψ2k of conservation laws (see [AI79]). These in turn give an in¬nite

double-sequence •2k+1 , •2k+1 of independent conservation laws for K = ’1, ex-

tending the two pairs that we already have. Although we will not discuss these,

it is worth pointing out that the generalized symmetries on (M (∞) , E (∞)) to

which they correspond under Noether™s theorem are not induced by symmetries

at any ¬nite prolongation. For this reason, they are called hidden symmetries.

Finally, we mention the following non-existence result, which is similar to the

result in Proposition 4.6 for higher-dimensional non-linear Poisson equations.

Proposition 4.8 In dimension n ≥ 3, there are no second-order Euclidean-

invariant conservation laws for the linear Weingarten system for hypersurfaces

in En+1 with Gauss curvature K = ’1.

Proof. In contrast to our proof of the analogous statement for non-linear

Poisson equations, we give here a direct argument not appealing to generating

functions. We work on the product

F (1) = F — {(a1, . . . , an) ∈ Rn : ai = ’1},

where F is the Euclidean frame bundle for En+1 , and the other factor param-

eterizes eigenvalues of admissible second fundamental forms. We use the usual

structure equations, but without the sum convention:

=’ ωij § ωj + πi § θ,

dωi

j

=’ πj § ω j

dθ

j

=’ ωik § ωkj + πi § πj ,

dωij

k

=’ πj § ωji.

dπi

j

There is a Pfa¬an system I on F (1) whose transverse n-dimensional integral

manifolds correspond to K = ’1 hypersurfaces; it is di¬erentially generated by

θ and the n 1-forms

θ i = π i ’ a i ωi .

A conservation law for this system is an (n ’ 1)-form on F (1) whose exterior

derivative vanishes on any integral manifold of I. A conservation law is invariant

under Euclidean motions if its restriction has the form

•= fi (a1 , . . . , an)ω(i).

What we will show is that for n ≥ 3, such a form cannot be closed modulo I

unless it equals 0.

190 CHAPTER 4. ADDITIONAL TOPICS

First, we calculate using the structure equations that

d• ≡ (dfi ’ fj ωij ) § ω(i) (mod I). (4.59)

i j

To proceed further, we want an expression for ωij § ω(i), which we obtain ¬rst

by computing

0≡ (mod I)

dθi

≡ ’ πk § ωki ’ dai § ωi + ai ωik § ωk

j k

≡ ’dai § ωi + (ai ’ ak )ωik § ωk .

k

and then by multiplying the last result by ω(ij):

0 ≡ dai § ω(j) + (ai ’ aj )ωij § ω(i) (mod I).

Using this in (4.59), we obtain

«

fi

dfi + daj § ω(i)

d• ≡ (mod I).

aj ’ a i

i j