erations, we will assume that HdR (M ) = 0 in all degrees q > 0. First, we have

the space gΠ, consisting of vector ¬elds on M which preserve I and Π,

gΠ = {v ∈ V(M ) : Lv I ⊆ I, Lv Π = 0}.

Second, we have the space of conservation laws

C = H n’1(„¦— (M )/EΛ );

under our topological assumption, this is identi¬ed with H n(EΛ ), and we need

not introduce a notion of “proper” conservation law as in §1.3. In this situation,

Noether™s theorem says the following.

5 But see Grassi, cit. p. 155n.

164 CHAPTER 4. ADDITIONAL TOPICS

There is a map · : gΠ ’ H n (EΛ), de¬ned by v ’ [v Π], which is

an isomorphism if Π is non-degenerate in a suitable sense.

The map is certainly well-de¬ned; that is, for any v ∈ gΠ , the form v Π is a

closed section of EΛ. First,

θ± )Ψ± ’ θ± § (v

v Π = (v Ψ± ),

Π is a section of EΛ ; and second,

so that v

Π) = Lv Π ’ v

d(v dΠ = 0,

so that v Π is closed. However, the proof that under the right conditions this

map is an isomorphism involves some rather sophisticated commutative algebra,

generalizing the symplectic linear algebra used in Chapter 1. This will not be

presented here.

As in the scalar case, a simple prescription for the conserved density in

(„¦— (M )/EΛ) corresponding to v ∈ gΠ is available when also

n’1

H

Lv Λ = 0.

One virtue of the Betounes form is that this holds for in¬nitesimal multi-contact

symmetries of [Λ]. Assuming only that dΛ = Π and Lv Λ = 0, we can calculate

that

Λ) = ’Lv Λ + v

d(’v dΛ = v Π. (4.31)

Therefore, ’v Λ ∈ „¦n’1(M ) represents a class in C = H n’1(„¦— (M )/EΛ)

corresponding to ·(v) ∈ H n (EΛ ). We will use this prescription in the following.

4.2.4 Harmonic Maps of Riemannian Manifolds

The most familiar variational PDE systems in di¬erential geometry are those

describing harmonic maps between Riemannian manifolds.

Let P, Q be Riemannian manifolds of dimensions n, s. We will de¬ne a La-

grangian density on P , depending on a map P ’ Q and its ¬rst derivatives,

whose integral over P may be thought of as the energy of the map. The appro-

priate multi-contact manifold for this is the space of 1-jets of maps P ’ Q,

M = J 1 (P, Q),

whose multi-contact system will be described shortly. We may also think of M

as Hom(T P, T Q), the total space of a rank-ns vector bundle over P — Q. To

carry out computations, it will be most convenient to work on

def

F = F(P ) — F(Q) — Rns,

4.2. EULER-LAGRANGE PDE SYSTEMS 165

where F(P ), F(Q) are the orthonormal frame bundles. These are parallelized

in the usual manner by (ω i , ωj ), (•± , •±), respectively, with structure equations

i

β

dωi = ’ωj § ωj ,

i

dωj = ’ωk § ωj + „¦i ,

i i k

j

γ

d• = ’•β § • , d•β = ’•γ § •β + ¦± .

± ± β ± ±

β

These forms and structure equations will be considered pulled back to F. To

complete a coframing of F, we take linear ¬ber coordinates p± on Rns, and

i

de¬ne

πi = dp± + •± pβ ’ p±ωi .

j

±

i βi j

The motivation here is that Hom(T P, T Q) ’ P —Q is a vector bundle associated

to the principal (O(n) — O(s))-bundle F(P ) — F(Q) ’ P — Q, with the data

((eP ), (eQ ), (p±)) ∈ F de¬ning the homomorphism eP ’ eQ p±. Furthermore,

± ±i

i i i

if a section σ ∈ “(Hom(T P, T Q)) is represented by an equivariant map (p± ) :

i

F(P ) — F(Q) ’ Rns, then the Rns-valued 1-form (πi ) represents the covariant

±

derivative of σ.

For our purposes, note that M = Hom(T P, T Q) is the quotient of F under

a certain action of O(n) — O(s), and that the forms semibasic for the projection

F ’ M are generated by ω i , •±, πi . A natural multi-contact system on M

±

pulls back to F as the Pfa¬an system I generated by

def

θ ± = •± ’ p ± ω i ,

i

and the associated integrable Pfa¬an system on M pulls back to J = {•±, ωi } =

{θ± , ωi }. The structure equations on F adapted to these Pfa¬an systems are

± ± ± i ± β

dθ = ’πi § ω ’ •β § θ ,

dωi = ’ωj § ωj ,

i

(4.32)

dπi = ¦± pβ ’ p±„¦j ’ •± § πi ’ πj § ωi .

β j

± ±

ji

βi β

We now de¬ne the energy Lagrangian

n

˜

Λ = 1 ||p||2ω ∈ “( J) ‚ „¦n(F),

2

where the norm is

||p||2 = Tr(p— p) = (p± )2.

i

˜

Although this Λ is not an admissible lifting of its induced functional [Λ], a

computation using the structure equations (4.32) shows that

def 1 2

+ p± θ± § ω(i)

2 ||p|| ω

Λ= i

is admissible:

’θ± § πi § ω(i) ’ p± pβ •± § ω + p±p± ωi § ω

j

±

dΛ = iiβ ij

’θ± § πi § ω(i),

±

=

166 CHAPTER 4. ADDITIONAL TOPICS

j

where the last step uses •± + •β = ωj + ωi = 0. Now we de¬ne

i

±

β

Π = ’θ± § πi § ω(i) ,

±

and note that Π is in fact the lift to F of a symmetric form on M , as de¬ned

earlier. Therefore, we have found the Betounes form and the Poincar´-Cartan

e

form for the energy functional.

The Euler-Lagrange system for [Λ], pulled back to F, is

EΛ = {θ± , πi § ωi , πi § ω(i) }.

± ±

A Legendre submanifold N ’ M = J 1 (P, Q) on which ωi = 0 is the 1-jet

graph of a map f : P ’ Q. On the inverse image π ’1(N ) ‚ F, in addition to

θ± = 0, there are relations

πi = h ± ω j ,

±

h± = h ± .

ij ij ji

Di¬erentiating this equation shows that the expression

h = h± ω i ω j — e Q

ij ±

is invariant along ¬bers of π ’1 (N ) ’ N , so it gives a well-de¬ned section

of Sym2(T — P ) — T Q; this is called the second fundamental form of the map

f : P ’ Q. The condition for N to be an integral manifold of the Euler-

Lagrange system is then

Tr(h) = h± = 0 ∈ “(M, f — T Q),

ii

De¬nition 4.9 A map f : P ’ Q between Riemannian manifolds is harmonic

if the trace of its second fundemental form vanishes.

Expressed in coordinates on P and Q, this is a second-order PDE system for

f : P ’ Q.

We now consider conservation laws for the harmonic map system EΛ ‚ „¦— (F)

corresponding to in¬nitesimal isometries (Killing vector ¬elds of either P or Q.

These are symmetries not only of Π but of the Lagrangian Λ, so we can use the

simpli¬ed prescription (4.31) for a conserved (n ’ 1)-form.

First, an in¬nitesimal isometry of P induces a unique vector ¬eld on M =

1

J (P, Q) preserving I and ¬xing Q. This vector ¬eld preserves Λ, Π, and EΛ ,

and has a natural lift to F which does the same. This vector ¬eld v on F

satis¬es

v ωi = vi , v •± = 0 ’ v θ± = ’p± vi , i

for some functions v i . We can then calculate

def

Λ ≡ ( 1 ||p||2vi ’ p± p±vj )ω(i) (mod {I}).

•v = v ij

2

4.2. EULER-LAGRANGE PDE SYSTEMS 167

As in Chapter 3, it is useful to write this expression restricted to the 1-jet graph

of a map f : P ’ Q, which is

= —P ( 1 ||p||2vi ’ p± p±vj )ωi

v Λ|N ij

2

( 1 ||df||2

= —P 1 v (ωi )2 ’ f — (•± )2 ) ,

2 2

where we use f — •± = p±ωi , and —P ω(i) = ωi . One might recognize the stress-

i

energy tensor

S = 1 ||df||2ds2 ’ f — ds2 ,

P Q

2

and write our conserved density as

Λ) ≡ —P (v (mod {I}).

2(v S) (4.33)

In fact, S is traditionally de¬ned as the unique symmetric 2-tensor on P for

which the preceding equation holds for arbitrary v ∈ V(P ) and f : P ’ Q;

then (4.33) gives a conserved density when v is an in¬nitesimal isometry and

f is a harmonic map. In this case In fact, for any in¬nitesimal isometry v, a

calculation gives

d(—P (v S)) = (v div S)ω (4.34)

on the 1-jet graph of any map.6

Now consider an in¬nitesimal isometry of Q, whose lift w ∈ V(F) satis¬es

ωi = 0, w •± = w i θ± = w±.

’

w w

Then

def

Λ = w±p± ω(i).

•w = w i

Given a map f : P ’ Q, we can use df ∈ Hom(T P, T Q) and ds2 ∈ Sym2(T — Q)

Q