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,