df = e1ω1 + ((cos „ )e2 + (sin „ )e3 )¯ 2 ,

¯ ω

and also

¯

df = d(f + re1 ) = e1 ω1 + e2 (ω2 + rω1 ) + e3 (rω1 ),

2 3

so that

±1

ω = ω1 ,

¯

(cos „ )¯ 2 = ω2 + rω1 ,

2

ω (4.66)

(sin „ )¯ 2 = rω1 .

3

ω

Note that ω2 +rω1 = (r cot „ )ω1 , and this gives a necessary condition on f alone

2 3

to share a pseudospherical line congruence. Similar calculations using e1 = e1 ¯

yield

¯2 2 3

ω1 = (cos „ )ω1 + (sin „ )ω1 ,

¯3 2 3

ω1 = ’(sin „ )ω1 + (cos „ )ω1 ,

and di¬erentiating the remaining relations (4.64) gives

¯3 3

ω2 = ω2 ,

¯

giving complete expressions for F — ω in terms of F — ω. Note in particular that

sin „ 2

¯3

ω1 = ω.

r

Now we can consider the curvature, expanding both sides of the de¬nition

¯¯

d¯ 1 = ’K ω1 § ω2.

ω2 ¯ (4.67)

4.3. HIGHER-ORDER CONSERVATION LAWS 195

First,

ω2 ω2 ¯ 3

’¯ 3 § ω1

d¯ 1 =

sin „ 2

2

’ω3 §

= ω

r

sin „ 1

ω § ω2 ,

= h21

r

where h21 is part of the second fundamental form of F : N ’ M , de¬ned by

ωi = hij ωj . Note in particular that if h12 = 0, then ω1 is a multiple of ω1 , so

3 3

by (4.66), ω 1 § ω2 = 0, a contradiction; we can now assume that h12 = h21 = 0.

¯ ¯

On the right-hand side of (4.67),

r

¯¯ ¯

’K ω 1 § ω 2 = ’Kω1 § ω3

¯

sin „ 1

r

¯ ω1 § ω2 .

= ’Kh12

sin „

Equating these expressions, we have

2

sin „

¯

K =’ 2

r

as claimed.

Now suppose given a surface f : N ’ E3 with constant negative Gauss

¯

curvature K = ’1. We are interested in ¬nding f which shares with f a

pseudospherical line congruence.

We start with local coordinates (s, t) on N whose coordinate lines ds = 0,

dt = 0 de¬ne the asymptotic curves of f. It will be convenient to instead have

orthogonal coordinate lines, so we de¬ne the coordinates x = s + t, y = s ’ t,

for which dx = 0 and dy = 0 de¬ne the principal curves of f. We have seen in

(4.55, 4.56) that the ¬rst and second fundamental forms are given by

= ds2 + 2 cos(2z)ds dt + dt2

I (4.68)

= cos2 z dx2 + sin2 z dy2 , (4.69)

II = 2 sin(2z)ds dt (4.70)

= sin(z) cos(z)(dx2 ’ dy2 ), (4.71)

where 2z is half of the angle measure between the asymptotic directions and

satis¬es the sine-Gordon equation. One orthonormal coframing is given by

(cos(z)dx, sin(z)dy); we consider an orthonormal coframing di¬ering from this

one by rotation by some ±:

ω1 cos ± sin ± cos(z)dx

= (4.72)

ω2 ’ sin ± cos ± sin(z)dy

cos(± ’ z) cos(± + z) ds

= . (4.73)

’ sin(± ’ z) ’ sin(± + z) dt

196 CHAPTER 4. ADDITIONAL TOPICS

The idea here is that we are looking for a function ± on N for which this

coframing could be part of that induced by a pseudospherical line congruence.

The main compatibility condition, derived from (4.66), is

ω2 + rω1 = r(cot „ )ω1 .

2 3

(4.74)

2

We can compute the Levi-Civita connection form ω1 using the structure equa-

tions dωi = ’ωj § ωj , and ¬nd

i

2

ω1 = (±s + zs )ds + (±t ’ zt )dt.

Similarly, we can compute from (4.70) and (4.73) the coe¬cients of the second

fundamental form with respect to the coframe (ω 1 , ω2 ), and ¬nd

3

ω1 = sin(± ’ z)ds ’ sin(± + z)dt,

3

ω2 = cos(± ’ z)ds ’ cos(± + z)dt.

Substituting these into the compatibility condition (4.74), we obtain an equation

of 1-forms whose ds, dt coe¬cients are

(csc „ + cot „ ) sin(± ’ z),

±s + z s =

1

±t ’ z t = sin(± + z).

csc „ + cot „

We compare this to (4.62), and conclude that the local existence of a solution

± is equivalent to having z satisfy the sine-Gordon equation. Note that the role

played by » in (4.62) is similar to that played by the angle „ in the pseudo-

spherical line congruence. We conclude by exhibiting a surprising use of the

B¨cklund transformation for the K = ’1 system. This starts with an integral

a

manifold in M 5 of E (see (4.49)) that is not transverse as a Legendre subman-

ifold, in the sense of being a 1-jet lift of an immersed surface in E3 . Instead,

regarding the contact manifold M as the unit sphere bundle over E3 , N ’ M

consists of the unit normal bundle of the line {(0, 0, w) : w ∈ R} ‚ E3 . This

Legendre surface is topologically a cylinder. To study its geometry, we will work

in the circle bundle F ’ M . Its preimage there is parameterized by

« «

e1 = (sin u cos v, ’ sin u sin v, cos u),

(u, v, w) ’ (0, 0, w), e2 = (cos u cos v, ’ cos u sin v, ’ sin u), , (4.75)

e3 = (sin v, cos v, 0)

where (u, v, w) ∈ S 1 —S 1 —R. It is easily veri¬ed that this is an integral manifold

for the pullback of E by F ’ M . We will apply the B¨cklund transformation

a

to this degenerate integral manifold, and obtain a non-trivial surface in E3 with

Gauss curvature K = ’1.

¯

We will take the B¨cklund transformation to be the submanifold B ‚ F — F

a

¯

(F is another copy of F) de¬ned by (4.63, 4.64, 4.65); this is a lift of the original

picture (4.61) from M to F. We ¬x the constants of the line congruence to be

π

„= , r = 1.

2

4.3. HIGHER-ORDER CONSERVATION LAWS 197

As a consequence, if our B¨cklund transformation gives a transverse Legendre

a

submanifold, then Theorem 4.3 states that the corresponding surface in E3 will

have Gauss curvature K = ’1.

¯

Now, the de¬nition of B ‚ F — F provides a unique lift π ’1 (N ) of our

¯

degenerate integral manifold (4.75) to B. The “other” K = ’1 system E should

¯

restrict to π’1(N ) ‚ B ‚ F — F to be algebraically generated by the 1-form

¯

θ, and then π’1 (N ) will be foliated into surfaces which project into integral

¯¯

¯

manifolds of E. So we compute θ = ω3 :

ω3

¯ = d¯, e3

x¯

dx + de1, ’e2

=

sin u dw ’ du.

=

Indeed, this 1-form is integrable, and its integral manifolds are of the form

u = 2 tan’1 (exp(w + c)), (4.76)

where c is an integration constant. We will consider the integral manifold cor-

responding to c = 0. The Euclidean surface that we are trying to construct is

now parameterized by x(u, v, w), constrained by (4.76). We obtain

¯

x(u, v, w) =

¯ x(u, v, w) + e1 (u, v, w)

(0, 0, w) + (sin u cos v, ’ sin u sin v, cos u)

=

2ew w

1’e2w

2e

cos v, ’ 1+e2w sin v, w +

= 1+e2w 1+e2w

(sech w cos v, ’sech w sin v, w ’ tanh w) .

=

This surface in E3 is the pseudosphere, the most familiar surface of constant

negative Gauss curvature; we introduced both it and the “framed line” in §1.4, as

examples of smooth but non-transverse Legendre submanifolds of the unit sphere

bundle M ’ E3 . In principle, we could iterate this B¨cklund transformation,

a

obtaining arbitrarily many examples of K = ’1 surfaces.

198 CHAPTER 4. ADDITIONAL TOPICS

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