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Proposition 4.7 On an immersed surface in E3 with constant Gauss curvature
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

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


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