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ВЇ
df = e1П‰1 + ((cos П„ )e2 + (sin П„ )e3 )ВЇ 2 ,
ВЇ П‰

and also
ВЇ
df = d(f + re1 ) = e1 П‰1 + e2 (П‰2 + rП‰1 ) + e3 (rП‰1 ),
2 3

so that
пЈ±1
пЈІ П‰ = П‰1 ,
ВЇ
(cos П„ )ВЇ 2 = П‰2 + rП‰1 ,
2
П‰ (4.66)
пЈі
(sin П„ )ВЇ 2 = rП‰1 .
3
П‰

Note that П‰2 +rП‰1 = (r cot П„ )П‰1 , and this gives a necessary condition on f alone
2 3

to share a pseudospherical line congruence. Similar calculations using e1 = e1 ВЇ
yield
ВЇ2 2 3
П‰1 = (cos П„ )П‰1 + (sin П„ )П‰1 ,
ВЇ3 2 3
П‰1 = в€’(sin П„ )П‰1 + (cos П„ )П‰1 ,
and diп¬Ђerentiating the remaining relations (4.64) gives

ВЇ3 3
П‰2 = П‰2 ,
ВЇ
giving complete expressions for F в€— П‰ in terms of F в€— П‰. Note in particular that
sin П„ 2
ВЇ3
П‰1 = П‰.
r
Now we can consider the curvature, expanding both sides of the deп¬Ѓnition
ВЇВЇ
dВЇ 1 = в€’K П‰1 в€§ П‰2.
П‰2 ВЇ (4.67)
4.3. HIGHER-ORDER CONSERVATION LAWS 195

First,

П‰2 П‰2 ВЇ 3
в€’ВЇ 3 в€§ П‰1
dВЇ 1 =
sin П„ 2
2
в€’П‰3 в€§
= П‰
r
sin П„ 1
П‰ в€§ П‰2 ,
= h21
r
where h21 is part of the second fundamental form of F : N в†’ M , deп¬Ѓned by
П‰i = hij П‰j . Note in particular that if h12 = 0, then П‰1 is a multiple of П‰1 , so
3 3

by (4.66), П‰ 1 в€§ П‰2 = 0, a contradiction; we can now assume that h12 = h21 = 0.
ВЇ ВЇ
On the right-hand side of (4.67),
r
ВЇВЇ ВЇ
в€’K П‰ 1 в€§ П‰ 2 = в€’KП‰1 в€§ П‰3
ВЇ
sin П„ 1
r
ВЇ П‰1 в€§ П‰2 .
= в€’Kh12
sin П„
Equating these expressions, we have
2
sin П„
ВЇ
K =в€’ 2
r
as claimed.

Now suppose given a surface f : N в†’ E3 with constant negative Gauss
ВЇ
curvature K = в€’1. We are interested in п¬Ѓnding 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 deп¬Ѓne the asymptotic curves of f. It will be convenient to instead have
orthogonal coordinate lines, so we deп¬Ѓne the coordinates x = s + t, y = s в€’ t,
for which dx = 0 and dy = 0 deп¬Ѓne the principal curves of f. We have seen in
(4.55, 4.56) that the п¬Ѓrst 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 в€’ dy2 ), (4.71)

where 2z is half of the angle measure between the asymptotic directions and
satisп¬Ѓes the sine-Gordon equation. One orthonormal coframing is given by
(cos(z)dx, sin(z)dy); we consider an orthonormal coframing diп¬Ђering from this
one by rotation by some О±:

П‰1 cos О± sin О± cos(z)dx
= (4.72)
П‰2 в€’ sin О± cos О± sin(z)dy
cos(О± в€’ z) cos(О± + z) ds
= . (4.73)
в€’ sin(О± в€’ z) в€’ sin(О± + z) dt
196 CHAPTER 4. ADDITIONAL TOPICS

The idea here is that we are looking for a function О± 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

П‰2 + rП‰1 = r(cot П„ )П‰1 .
2 3
(4.74)
2
We can compute the Levi-Civita connection form П‰1 using the structure equa-
tions dП‰i = в€’П‰j в€§ П‰j , and п¬Ѓnd
i

2
П‰1 = (О±s + zs )ds + (О±t в€’ zt )dt.

Similarly, we can compute from (4.70) and (4.73) the coeп¬ѓcients of the second
fundamental form with respect to the coframe (П‰ 1 , П‰2 ), and п¬Ѓnd
3
П‰1 = sin(О± в€’ z)ds в€’ sin(О± + z)dt,
3
П‰2 = cos(О± в€’ z)ds в€’ cos(О± + z)dt.

Substituting these into the compatibility condition (4.74), we obtain an equation
of 1-forms whose ds, dt coeп¬ѓcients are

(csc П„ + cot П„ ) sin(О± в€’ z),
О±s + z s =
1
О±t в€’ z t = sin(О± + z).
csc П„ + cot П„
We compare this to (4.62), and conclude that the local existence of a solution
О± is equivalent to having z satisfy the sine-Gordon equation. Note that the role
played by О» in (4.62) is similar to that played by the angle П„ in the pseudo-
spherical line congruence. We conclude by exhibiting a surprising use of the
BВЁcklund transformation for the K = в€’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 в†’ M
consists of the unit normal bundle of the line {(0, 0, w) : w в€€ R} вЉ‚ E3 . This
Legendre surface is topologically a cylinder. To study its geometry, we will work
in the circle bundle F в†’ M . Its preimage there is parameterized by
пЈ« пЈ« пЈ¶пЈ¶
e1 = (sin u cos v, в€’ sin u sin v, cos u),
(u, v, w) в†’ пЈ­(0, 0, w), пЈ­ e2 = (cos u cos v, в€’ cos u sin v, в€’ sin u), пЈёпЈё , (4.75)
e3 = (sin v, cos v, 0)

where (u, v, w) в€€ S 1 Г—S 1 Г—R. It is easily veriп¬Ѓed that this is an integral manifold
for the pullback of E by F в†’ M . We will apply the BВЁcklund transformation
a
to this degenerate integral manifold, and obtain a non-trivial surface in E3 with
Gauss curvature K = в€’1.
ВЇ
We will take the BВЁcklund transformation to be the submanifold B вЉ‚ F Г— F
a
ВЇ
(F is another copy of F) deп¬Ѓned by (4.63, 4.64, 4.65); this is a lift of the original
picture (4.61) from M to F. We п¬Ѓx the constants of the line congruence to be
ПЂ
П„= , r = 1.
2
4.3. HIGHER-ORDER CONSERVATION LAWS 197

As a consequence, if our BВЁcklund transformation gives a transverse Legendre
a
submanifold, then Theorem 4.3 states that the corresponding surface in E3 will
have Gauss curvature K = в€’1.
ВЇ
Now, the deп¬Ѓnition of B вЉ‚ F Г— F provides a unique lift ПЂ в€’1 (N ) of our
ВЇ
degenerate integral manifold (4.75) to B. The вЂњotherвЂќ K = в€’1 system E should
ВЇ
restrict to ПЂв€’1(N ) вЉ‚ B вЉ‚ F Г— F to be algebraically generated by the 1-form
ВЇ
Оё, and then ПЂв€’1 (N ) will be foliated into surfaces which project into integral
ВЇВЇ
ВЇ
manifolds of E. So we compute Оё = П‰3 :

П‰3
ВЇ = dВЇ, e3
xВЇ
dx + de1, в€’e2
=
sin u dw в€’ du.
=

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

u = 2 tanв€’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
ВЇ

x(u, v, w) =
ВЇ x(u, v, w) + e1 (u, v, w)
(0, 0, w) + (sin u cos v, в€’ sin u sin v, cos u)
=
2ew w
1в€’e2w
2e
cos v, в€’ 1+e2w sin v, w +
= 1+e2w 1+e2w

(sech w cos v, в€’sech w sin v, w в€’ tanh w) .
=

This surface in E3 is the pseudosphere, the most familiar surface of constant
negative Gauss curvature; we introduced both it and the вЂњframed lineвЂќ in В§1.4, as
examples of smooth but non-transverse Legendre submanifolds of the unit sphere
bundle M в†’ E3 . In principle, we could iterate this BВЁcklund transformation,
a
obtaining arbitrarily many examples of K = в€’1 surfaces.
198 CHAPTER 4. ADDITIONAL TOPICS
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