<< Предыдущая стр. 43(из 48 стр.)ОГЛАВЛЕНИЕ Следующая >>
g(xi , p(k)) = h(x, p(kв€’2)) + hK (x, p(kв€’2))pK + hI (x, p(kв€’3))pI .

Again working modulo Rkв€’1, the only term that is non-linear in (pK , pI ) is
в€‚hK K
so as before, we can assume в€‚hA = 0 and write
в€‚pA pAi pKi , в€‚p

g(xi , p(k)) = h(x, p(kв€’2)) + hK (x, p(kв€’3))pK + hI (x, p(kв€’3))pI .

The п¬Ѓnal step is similar, and gives that h is linear in pA . This yields

g(xi , p(k)) = h + hApA + hK pK + hI pI , (4.48)
182 CHAPTER 4. ADDITIONAL TOPICS

where each of h, hA, hK , hI is a function of (xi , p(kв€’3)). This is the п¬Ѓrst state-
ment of the proposition.
To derive the second statement, we use the form (4.48) in the condition
(4.46) modulo Rkв€’2, in which the only term non-linear in (pK , pI ) is

d|I|f
I i (kв€’3)
0в‰Ў h (x , p )I
dx
hI (xi , p(kв€’3))f (p)
в‰Ў pI\i pi.
iв€€I

In particular, if f (p) = 0, then we must have hI = 0. Therefore g в€€ Rk
actually lies in Rkв€’1, and we can induct downward on k, eventually proving
that g в€€ R1 , as desired.

We mention two situations which contrast sharply with that of the non-linear
Poisson equation in dimension n в‰Ґ 3. First, in the case of a linear Poisson
equation в€†z = f(z), f (z) = 0 (still in n в‰Ґ 3 dimensions), one can extend the
preceding argument to show that a generating function for a conservation law
is linear in all of the derivative variables pI . In particular, the inп¬Ѓnite collection
of conservation laws for the Laplace equation в€†z = 0 can be determined in this
manner; it is interesting to see how all of these disappear upon the addition of
a non-linear term to the equation.
Second, in dimension n = 2, there are well-known non-linear Poisson equa-
tions в€†u = sinh u and в€†u = eu having inп¬Ѓnitely many higher-order conservation
laws, but we will not discuss these.

The K = в€’1 Surface System
4.3.3
In В§1.4, we constructed Monge-Ampere systems on the contact manifold M 5
of oriented tangent planes to Euclidean space E3 , whose integral manifolds
corresponded to linear Weingarten surfaces. We brieп¬‚y recall this setup for
the case of surfaces with Gauss curvature K = в€’1. Our index ranges are now
1 в‰¤ a, b, c в‰¤ 3, 1 в‰¤ i, j, k в‰¤ 2.
Let F в†’ E3 be the Euclidean frame bundle, with its global coframing П‰ a ,
a b
П‰b = в€’П‰a satisfying the structure equations

dП‰a = в€’П‰b в€§ П‰b ,
a a a c
dП‰b = в€’П‰c в€§ П‰b .

We set Оё = П‰3 , ПЂi = П‰i , and then Оё в€€ в„¦1(F) is the pullback of a global contact
3

form on M = G2 (T E3 ). The forms that are semibasic over M are generated by
Оё, П‰i , ПЂi в€€ в„¦1 (F).
We deп¬Ѓne the 2-forms on F
О˜ = dОё = в€’ПЂ1 в€§ П‰1 в€’ ПЂ2 в€§ П‰2 ,
ОЁ = ПЂ 1 в€§ ПЂ2 + П‰ 1 в€§ П‰ 2 ,
which are pullbacks of uniquely determined forms on M . On a transverse inte-
gral element E 2 вЉ‚ Tm M of the contact system I = {Оё, О˜}, on which П‰ 1 в€§П‰2 = 0,
4.3. HIGHER-ORDER CONSERVATION LAWS 183

there are relations
ПЂi = hij П‰j , hij = hji.
In this case,
ПЂ1 в€§ ПЂ2 = KП‰1 в€§ П‰2 ,
where K = h11h22 в€’ h12h21 is the Gauss curvature of any surface N 2 в†’ E3
whose 1-jet graph in M 5 is tangent to E вЉ‚ Tm M . Therefore, transverse integral
manifolds of the EDS

E = {Оё, О˜, ОЁ} (4.49)

correspond locally to surfaces in E3 with constant Gauss curvature K = в€’1.
The EDS (M, E) is an example of a hyperbolic Monge-Ampere system; this
notion appeared in В§2.1, where we used it to specify a branch of the equiva-
lence problem for PoincarВґ-Cartan forms on contact 5-manifolds. The deп¬Ѓning
e
property of a hyperbolic Monge-Ampere system E = {Оё, О˜, ОЁ} is that modulo
the algebraic ideal {Оё}, E contains two distinct (modulo scaling) decomposable
2-forms; that is, one can п¬Ѓnd two non-trivial linear combinations of the form

О± 1 в€§ ОІ1 ,
О»1 О˜ + Вµ 1 ОЁ =
О± 2 в€§ ОІ2 .
О»2 О˜ + Вµ 2 ОЁ =

This exhibits two rank-2 Pfaп¬ѓan systems Ii = {О±i, ОІi }, called the characteristic
systems of E, which are easily seen to be independent of choices (except for which
one is I1 and which one is I2 ). The relationship between the geometry of the
characteristic systems and that of the original hyperbolic Monge-Ampere system
is very rich (see [BGH95]). Of particular interest are those hyperbolic systems
whose characteristic systems each contain a non-trivial conservation law. We
will show that this holds for the K = в€’1 system introduced above, but only
after one prolongation. In other words, for the prolonged system E (1) , there is
(1)
also a notion of characteristic systems Ii which restrict to any integral surface
(1)
as the original Ii , and each of these Ii contains a non-trivial conservation law
for E (1).
Returning to the discussion of integral elements of E = {Оё, О˜, ОЁ}, note that
for any integral element E вЉ‚ T(p,H) M , given by equations

ПЂi в€’ hij П‰j = 0,

there is a unique frame (p, (e1 , e2, e1 Г— e2 )) в€€ F over (p, e1 в€§ e2 ) в€€ M for which
the second fundamental form is normalized as
1
h22 = в€’ a ,
h11 = a > 0, h12 = h21 = 0.

The tangent lines in E3 spanned by these e1 , e2 are the principal directions at
p of any surface whose 1-jet graph is tangent to E; they deп¬Ѓne an orthonormal
frame in which the second fundamental form is diagonal. The number a > 0 is
184 CHAPTER 4. ADDITIONAL TOPICS

determined by the plane E вЉ‚ T M , so to study integral elements of (M, E), and
in particular to calculate on its п¬Ѓrst prolongation, we introduce

F (1) = F Г— Rв€—,

where Rв€— has the coordinate a > 0. There is a projection F (1) в†’ M (1) , mapping

(p, e, a) в†’ (p, e1 в€§ e2 , {ПЂ1 в€’ aП‰1 , ПЂ2 + a П‰2 }вЉҐ ).
1

We deп¬Ѓne on F (1) the forms
Оё1 = ПЂ1 в€’ aП‰1 ,
Оё2 = ПЂ 2 + a П‰ 2 ,
1

which are semibasic for F (1) в†’ M (1). The п¬Ѓrst prolongation of the system E
on M is a Pfaп¬ѓan system on M (1) , which pulls back to F (1) as

E (1) = {Оё, Оё1 , Оё2, dОё1, dОё2 }.

We have structure equations
пЈј
dОё в‰Ў в€’Оё1 в€§ П‰1 в€’ Оё2 в€§ П‰2 в‰Ў 0 пЈЅ
2
dОё1 в‰Ў в€’da в€§ П‰1 + 1+a П‰2 в€§ П‰2
1
(mod {Оё, Оё1 , Оё2}),
a2 пЈѕ
dОё2 в‰Ў a2 da в€§ П‰2 + 1+a П‰2 в€§ П‰1
1
1
a

and in particular, we have the decomposable linear combinations

в€’dОё1 в€’ a dОё2 = (da в€’ (1 + a2 )П‰2 ) в€§ (П‰1 + a П‰2 ),
1
1
(4.50)
в€’dОё1 + a dОё2 = (da + (1 + a2 )П‰2 ) в€§ (П‰1 в€’ a П‰2 ).
1
1

The EDS E (1) is algebraically generated by Оё, Оё1 , Оё2 , and these two decomposable
2-forms. The characteristic systems are by deп¬Ѓnition diп¬Ђerentially generated by
(1)
I1 = {Оё, Оё1 , Оё2 , da в€’ (1 + a2 )П‰2 , П‰1 + a П‰2 },
1
1
(4.51)
(1)
I2 = {Оё, Оё1 , Оё2 , da + (1 + a2 )П‰2 , П‰1 в€’ a П‰2 }.
1
1

Now, the вЂњuniversalвЂќ second fundamental form can be factored as

II = a(П‰1 )2 в€’ a (П‰2 )2 = a(П‰1 + a П‰2 )(П‰1 в€’ 1 П‰2 ).
1 1
(4.52)
a

These linear factors, restricted any K = в€’1 surface, deп¬Ѓne the asymptotic curves
of that surface, so by comparing (4.51) and (4.52) we п¬Ѓnd that:
On a K = в€’1 surface, the integral curves of the characteristic sys-
tems are the asymptotic curves.
(1)
Now we look for Euclidean-invariant conservation laws in each Ii . Instead
of using NoetherвЂ™s theorem, we work directly. We start by setting
(1)
П•1 = f(a)(П‰1 + a П‰2 ) в€€ I1 ,
1
4.3. HIGHER-ORDER CONSERVATION LAWS 185

and seek conditions on f(a) to have dП•1 в€€ E (1). A short computation using the
structure equations gives

dП•1 в‰Ў (f (a)(1 + a2) в€’ f(a)a)П‰2 в€§ (П‰1 + a П‰2 )
1
(mod E (1)),
1

so the condition for П•1 to be a conserved 1-form is

f (a) a
= .
1 + a2
f(a)

A solution is
1 + a2 (П‰1 + 1 П‰2);
1
П•1 = 2 a
1
the choice of multiplicative constant 2 will simplify later computations. A
similar computation, seeking an appropriate multiple of П‰ 1 в€’ a П‰2 , yields the
1

conserved 1-form
П•2 = 1 1 + a2 (П‰1 в€’ 1 П‰2).
2 a

On any simply connected integral surface of the K = в€’1 system, there are
coordinate functions s, t such that

П•1 = ds, П•2 = dt.

If we work in these coordinates, and in particular use the non-orthonormal
coframing (П•1 , П•2) then we can write

1
П‰1 в€љ
= (П•1 + П•2 ), (4.53)
1 + a2
a
П‰2 в€љ (П•1 в€’ П•2 ),
= (4.54)
1 + a2
1 в€’ a2
12 22 2
П•1 П•2 + (П•2 )2 ,
I = (П‰ ) + (П‰ ) = (П•1 ) + 2 (4.55)
1 + a2
4a
II = П•1 П•2 . (4.56)
1 + a2

These expressions suggest that we deп¬Ѓne

a = tan z,

where a > 0 means that we can smoothly choose z = tanв€’1 a в€€ (0, ПЂ/2). Note
that 2z is the angle measure between the asymptotic directions П•вЉҐ , П•вЉҐ , and
1 2
that

П‰1 = (cos z)(П•1 + П•2 ), П‰2 = (sin z)(П•1 в€’ П•2 ). (4.57)

The following is fundamental in the study of K = в€’1 surfaces.
186 CHAPTER 4. ADDITIONAL TOPICS

 << Предыдущая стр. 43(из 48 стр.)ОГЛАВЛЕНИЕ Следующая >>