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. 45
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So • is a conservation law only if for each i,

daj
dfi = fi .
ai ’ a j
j


Keep in mind that we are requiring this equation to hold on the locus F (1)
where Πai = ’1. Wherever at least one fi (a1 , . . . , an) is non-zero, we must
have « 
dai § daj
daj 
0 = d =’ .
(ai ’ aj )2
ai ’ a j
j j

However, when n ≥ 3, the summands in the expression are linearly independent
2-forms on F (1) .

It is worth noting that when n = 2, this 2-form does vanish, and we can
solve for f1 (a1 , a2), f2 (a1, a2) to obtain the conservation laws •1 , •2 discussed
earlier. The same elementary method can be used to analyze second-order
conservation laws for more general Weingarten equations; in this way, one can
obtain a full classi¬cation of those few Wiengarten equations possessing higher-
order conservation laws.
4.3. HIGHER-ORDER CONSERVATION LAWS 191

4.3.4 Two B¨cklund Transformations
a
We have seen a relationship between the K = ’1 surface system, and the sine-
Gordon equation
1
zxy = sin(2z). (4.60)
2

Namely, the half-angle measure between the asymptotic directions on a K = ’1
surface, when expressed in asymptotic coordinates, satis¬es the sine-Gordon
equation. We have also interpreted this relationship in terms of important EDS
constructions. In this section, we will explain how this relationship connects the
B¨cklund transformations associated to each of these systems.
a
There are many de¬nitions of B¨cklund transformation in the literature, and
a
instead of trying to give an all-encompassing de¬nition, we will restrict attention
to Monge-Ampere systems
E = {θ, ˜, Ψ},
where θ is a contact form on a manifold (M 5 , I), and ˜, Ψ ∈ „¦2(M ) are linearly
¯¯
independent modulo {I}. Suppose that (M, E) and (M , E) are two Monge-
Ampere systems, with
¯ ¯¯ ¯
E = {θ, ˜, Ψ}, E = {θ, ˜, Ψ}.
¯¯
A B¨cklund transformation between (M, E) and (M , E), is a 6-dimensional sub-
a
¯
manifold B ‚ M — M such that in the diagram

(4.61)
BA
˜ AA
π ˜˜ π
¯
˜ AA
˜˜ AA
˜˜
¯
M,
M

¯
• each projection B ’ M , B ’ M is a submersion; and
• pulled back to B, we have
¯ ¯ ¯
{Ψ, Ψ} ≡ {˜, ˜} (mod {θ, θ}).

The second condition implies that the dimension of the space of 2-forms spanned
¯¯ ¯
by {˜, Ψ, ˜, Ψ} modulo {θ, θ} is at most 2. Therefore,
¯¯ ¯
{˜, Ψ} ≡ {˜, Ψ} (mod {θ, θ}).

This consequence is what we really want, but the original formulation has the
¯
extra bene¬t of ruling out linear dependence between ˜ and ˜, which would
lead to a triviality in what follows.
A B¨cklund transformation allows one to ¬nd a family of integral manifolds
a
¯¯
of (M , E) from one integral manifold N 2 ’ M of (M, E), as follows. On the
¯¯
3-dimensional preimage π ’1 (N ) ‚ B, the restriction π — E is algebraically gen-
¯
erated by θ alone, and is therefore an integrable Pfa¬an system. Its integral
192 CHAPTER 4. ADDITIONAL TOPICS

manifolds can therefore be found by ODE methods, and they foliate π ’1(N )
into a 1-parameter family of surfaces which project by π to integral manifolds
¯
¯ , E). In each of the following two examples, (M, E) and (M , E) are equal,
¯ ¯¯
of (M
so one can generate from one known solution many others.

Example 1: B¨cklund transformation for the sine-Gordon equation.
a
The primary example concerns the sine-Gordon equation (4.60). The well-
known coordinate phenomenon is that if two functions u(x, y), u(x, y) satisfy
¯
the ¬rst-order PDE system
ux ’ ux = » sin(u + u),
¯ ¯
(4.62)
1
uy + uy = » sin(u ’ u),
¯ ¯

where » = 0 is any constant,11 then each of u(x, y) and u(x, y) satis¬es (4.60).
¯
Conversely, given a function u(x, y), the overdetermined system (4.62) for un-
¯
known u(x, y) is compatible, and can therefore be reduced to an ODE system,
if and only if u(x, y) satis¬es (4.60). This indicates that given one solution of
¯
the sine-Gordon equation, ODE methods give a family of additional solutions.
We ¬t this example into our de¬nition of a B¨cklund transformation as
a
follows. Start with two copies of the sine-Gordon Monge-Ampere system, one
on M = {(x, y, u, p, q)} generated by
± 
θ = du ’ p dx ’ q dy,
 
˜ = dθ = ’dp § dx ’ dq § dy,
E= ,
 
Ψ = dx § dp + dq § dy ’ sin(2u)dx § dy
¯
the other on M = {(¯, y, u, p, q)} generated by
x¯¯¯¯
± 
¯
 θ = d¯ ’ p d¯ ’ q d¯,
u ¯x ¯y 
¯ = dθ = ’d¯ § d¯ ’ d¯ § d¯,
¯
¯=
E ˜ p x q y .
¯ 
Ψ = d¯ § d¯ + d¯ § d¯ ’ sin(2¯)d¯ § d¯
x p q y ux y
¯
One can verify that the submanifold B ‚ M — M de¬ned by
±
 x = x, y = y,
¯ ¯
p ’ p = » sin(u + u),
¯ ¯
 1
q + q = » sin(u ’ u),
¯ ¯
satis¬es the criteria for a B¨cklund transformation, and that the process of
a
solving the overdetermined system (4.62) for u(x, y) corresponds to integrating
the Frobenius system as described previously.
For example, the solution u(x, y) = 0 of sine-Gordon corresponds to the
¯
¯
integral manifold N = {(x, y, 0, 0, 0)} ‚ M , whose preimage in B ‚ M — M has
coordinates (x, y, u) and satis¬es
1
u = p = q = 0, p = » sin(u), q =
¯¯¯ sin(u).
»
11 This » will not correspond to the integration parameter in the B¨cklund transformation.
a
It plays a role only in the relation to the K = ’1 system, to be discussed shortly.
4.3. HIGHER-ORDER CONSERVATION LAWS 193

The system E is algebraically generated by the form
1
θ|π’1 (N ) = du ’ » sin(u)dx ’ sin(u)dy.
¯ »

The problem of ¬nding u(x, y) on which this θ vanishes is the same as solving
the overdetermined system (4.62) with u = 0. It is obtained by integrating
¯

du 1
’ » dx ’ dy = 0,
»
sin u
which has the implicit solution
1
’ ln(csc u + cot u) ’ »x ’ » y = c,

where c is the integration constant. This can be solved for u to obtain
1
u(x, y) = 2 tan’1 (e»x+ » y+c ).

One can verify that this is indeed a solution to the sine-Gordon equation. In
principle, we could rename this as u, and repeat the process to obtain more
¯
solutions.

Example 2: B¨cklund transformation for the K = ’1 system.
a
¯
Suppose that f, f : N ’ E3 are two immersions of a surface into Euclidean
¯
space. We say that there is a pseudospherical line congruence between f, f if
for each p ∈ N :
¯
1. the line through f(p) and f (p) in E3 is tangent to each surface at these
¯
points (we assume f(p) = f (p));
¯
2. the distance r = ||f(p) ’ f (p)|| is constant;
3. the angle „ between the normals ν(p) and ν (p) is constant.
¯
¯
This relationship between f, f will play a role analogous to that of the system
(4.62). We prove the following theorem of Bianchi.
¯
Theorem 4.3 If there is a pseudospherical line congruence between f, f : N ’
¯
E3 , then each of f and f has constant negative Gauss curvature

sin2 („ )
K=’ .
r2


¯
It is also true that given one surface f , there locally exists a surface f sharing
¯ ¯
a pseudospherical line congruence with f if and only if f has constant nega-
tive Gauss curvature. We will partly verify this claim, after proving Bianchi™s
theorem.
194 CHAPTER 4. ADDITIONAL TOPICS

¯
Proof. Choose Euclidean frame ¬elds F, F : N ’ F which are adapted to the
pair of surfaces in the sense that

e1 = e1 ,
¯ (4.63)

made possible by condition 1 above. Also, as usual, we let e3 , e3 be unit normals
¯
¯, respectively, which must then satisfy
to f, f

e2 = (cos „ )e2 + (sin „ )e3 ,
¯
(4.64)
e3 = (’ sin „ )e2 + (cos „ )e3 ,
¯

with „ constant by condition 3. Now condition 2 says that
¯
f (p) = f(p) + re1 (p) (4.65)

for ¬xed r. We can use the structure equations
j
df = ei · ωi , dei = ej · ωi ,
¯e ¯
and similar for df, d¯i, to obtain relations among the pullbacks by F and F of
the canonical forms on F. Namely,

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. 45
( 48 .)



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