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¯
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

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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