Biharmonic maps into compact Lie groups and the integrable systems

0.17 MB·English

by  Hajime Urakawa

BIHARMONIC MAPS INTO COMPACT LIE GROUPS
AND THE INTEGRABLE SYSTEMS
9
HAJIME URAKAWA
0
0
2

t
c
O Abstract. Inthis paper,the reductionofbiharmonic mapequa-
tion in terms of the Maurer-Cartanform for all smooth map of an
5
arbitrarycompactRiemannianmanifoldinto a compactLie group

(G,h) with bi-invariant Riemannian metric h is obtained. By this
]
G formula, all biharmonic curves into compaqct Lie groups are de-
D termined, and all the biharmonic maps of an open domain of R2
. withthe conformalmetricofthe standardRiemannianmetricinto
h
(G,h) are determined.
t
a
m
[ 1. Introduction and statement of results.

1
The theory of harmonic maps into Lie groups, symmetric spaces or
v
2 homogeneous spaces has been extensively studied related to the inte-
9
grablesystems by many mathematicians (for examples, [6], [7], [3], [2]).
6
0 In this paper, we want to build a bridge between the theory of bihar-
. monic maps into Lie groups and the integrable systems extending the
0
1 above harmonic map theory.
9
0
: Acknowledgement: The author expresses his gratitude to Prof. J.
v
i Inoguchi who gave many useful suggestions and Prof. A. Kasue for his
X
financial support during the preparation of this paper .
r
a
2. Preliminaries.
In this section, we prepare general materials and facts harmonic
maps, biharmonic maps into compact Lie groups (cf. [5]).
Let (M,g) be an m dimensional compact Riemannian manifold, and
G be an n dimensional compact Lie group with Lie algebra g, and h,
2000 Mathematics Subject Classification. 58E20.
Key words and phrases. harmonic map, biharmonic map, compact Lie group,
integrable system, Maurer-Cartanform.
Supported by the Grant-in-Aid for the Scientific Research, (A), No. 19204004;
(C), No. 21540207,Japan Society for the Promotion of Science.
1
2 HAJIME URAKAWA
the bi-invariant Riemannian metric on G corresponding to the Ad(G)-
invariant inner product , on g.
h i
The energy functional on the space C (M,G) of all C maps of M
∞ ∞
into G is defined by
1
E(ψ) = dψ 2v ,
g
2 M k k
Z
and for a C one parameter deformation ψ C (M,G) ( ǫ < t < ǫ)
∞ t ∞
∈ −
of ψ with ψ = ψ, the first variation formula is given by
0
d
E(ψ ) = τ(ψ),V v ,
t g
dt(cid:12) − Mh i
(cid:12)t=0 Z
(cid:12)
where V is a variatio(cid:12)(cid:12)n vector field along ψ defined by V = ddt t=0ψt
which belongs to the space Γ(ψ 1TG) of sections of the induced b(cid:12)undle
− (cid:12)
of the tangent bundle TG by ψ. The tension field τ(ψ) is defined(cid:12) by
m
τ(ψ) = B(ψ)(e ,e ), (2.1)
i i
i=1
X
where
B(ψ)(X,Y) = h dψ(Y) dψ( Y)
∇dψ(X) − ∇X
for X,Y X(M). Here, , and h, are the Levi-Civita connections
∈ ∇ ∇
of (M,g) and (G,h), respectively. For a harmonic map ψ : (M,g)
→
(G,h), the second variation formula of the energy functional E(ψ) is
d2
E(ψ ) = J(V),V g
t g
dt2(cid:12) Mh i
(cid:12)t=0 Z
(cid:12)
where (cid:12)
(cid:12)
J(V) = ∆V (V),
−R
m
∆V = ∇∗∇V = − {∇ei(∇eiV)−∇∇eieiV},
i=1
X
m
(V) = Rh(V,dψ(e ))dψ(e ).
i i
R
i=1
X
Here, is the induced connection on the induced bundle ψ 1TG,
−
∇
and is Rh is the curvature tensor of (G,h) given by Rh(U,V)W =
[ h, h]W h W (U,V,W X(G)).
∇U ∇V −∇[U,V] ∈
The bienergy functional is defined by
1 1
E (ψ) = (d+δ)2ψ 2v = τ(ψ) 2v , (2.2)
2 g g
2 M k k 2 M | |
Z Z
BIHARMONIC MAPS INTO COMPACT LIE GROUPS 3
and the first variation formula of the bienergy is given (cf. [4]) by
d
E (ψ ) = τ (ψ),V v (2.3)
2 t 2 g
dt(cid:12) − Mh i
(cid:12)t=0 Z
(cid:12)
where the bitension fi(cid:12)eld τ (ψ) is defined by
(cid:12) 2
τ (ψ) = J(τ(ψ)) = ∆τ(ψ) (τ(ψ)), (2.4)
2
−R
and a C map ψ : (M,g) (G,h) is called to be biharmonic if
∞
→
τ (ψ) = 0. (2.5)
2
3. Determination of the bitension field
Now, let θ be the Maurer-Cartan form on G, i.e., a g-valued left
invariant 1-form on G which is defined by θ (Z ) = Z, (y G, Z g).
y y
∈ ∈
For every C map ψ of (M,g) into (G,h), let us consider a g-valued
∞
1-form α on M given by α = ψ θ. Then, it is well known (see for
∗
example, [2]) that
Lemma 3.1. For every C map ψ : (M,g) (G,h),
∞
→
θ(τ(ψ)) = δα (3.1)
−
where α = ψ θ, and θ is the Maurer-Cartan form of G.
∗
Thus, ψ : (M,g) (G,h) is harmonic if and only if δα = 0.
→
Furthemore, let X n be an orthonormal basis of g with respect
{ s}s=1
to the inner product , . Then, for every V Γ(ψ 1TG),
−
h i ∈
n
V(x) = h (V(x),X )X T G,
ψ(x) sψ(x) sψ(x) ψ(x)
∈
i=1
X
n
θ(V)(x) = h (V(x),X )X g, (3.2)
ψ(x) sψ(x) s
∈
i=1
X
for all x M. Then, for every X X(M),
∈ ∈
n
θ( V) = h( V,X )X
X X s s
∇ ∇
s=1
X
n
= Xh(V,X )X h(V, X ) X
s s X s s
{ − ∇ }
s=1
X
n
= X(θ(V)) h(V, X )X , (3.3)
X s s
− ∇
s=1
X
4 HAJIME URAKAWA
whereweregardedavectorfieldY X(G)tobeanelement inΓ 1(TG)
−
∈
by
Y(x) = Y(ψ(x)), (x M),
∈
and we useed the compatibility condition:
Xh(V,W) = h( V,W)+h(V, W), (V,W Γ(ψ 1TG)),
X X −
∇ ∇ ∈
for all X X(M).
∈
For every X X(M), the differential of ψ, ψ X Γ(ψ 1TG) which
−
∈ ∗ ∈
is given by ψ X T G (x M), can be written as
x ψ(x)
∗ ∈ ∈
n
ψ X = h(ψ X ,X )X , (x M),
x x tψ(x) tψ(x)
∗ ∗ ∈
t=1
X
which yields that
n
( X ) = h(ψ X ,X )( h X ) (3.4)
∇X s x ∗ x tψ(x) ∇Xt s ψ(x)
t=1
X
for all x M.
∈
Here, let us recall that the Levi-Civita connection h of (G,h) is
∇
given (cf. [5] Vol. II, p. 201, Theorem 3.3) by
1 1 n
h X = [X ,X ] = Cℓ X , (3.5)
∇Xt s 2 t s 2 ts ℓ
ℓ=1
X
where the structure constant Cℓ of g is defined by
ts
n
[X ,X ] = Cℓ X ,
t s ts ℓ
ℓ=1
X
and satisfies that
Cℓ = [X ,X ],X
ts h t s ℓi
= X ,[X ,X ]
s t ℓ
−h i
= Cs. (3.6)
− tℓ
BIHARMONIC MAPS INTO COMPACT LIE GROUPS 5
Thus, we have by (3.4) and (3.5),
n 1 n n
h(V, X )X = h V, h(ψ X,X )Cℓ X X
∇X s s 2 ∗ t ts ℓ! s
s=1 s,t=1 ℓ=1
X X X
1 n
= h(V,X )h(ψ X,X )CsX
−2 ℓ ∗ t tℓ s
s,t,ℓ=1
X
1 n
= h(V,X )h(ψ X,X )[X ,X ]
ℓ t t ℓ
−2 ∗
t,ℓ=1
X
1 n n
= h(ψ X,X )X , h(V,X )X
t t ℓ ℓ
−2 " ∗ #
t=1 ℓ=1
X X
1
= [α(X),θ(V)], (3.7)
−2
because we have
α(X) = (ψ θ)(X)
∗
= θ(ψ X)
∗
n
= θ h(ψ X,X )X
t t
∗ !
t=1
X
n
= h(ψ X,X )X , (3.8)
t t
∗
t=1
X
and
n
θ(V) = h(V,X )θ(X )
ℓ ℓ
ℓ=1
X
n
= h(V,X )X . (3.9)
ℓ ℓ
ℓ=1
X
Substituting (3.8) and (3.9) into the above, we have the last equality
of (3.7).
Therefore, we obtain the following lemma together with (3.7) and
(3.3).
Lemma 3.2. For every C map ψ : (M,g) (G,h),
∞
→
1
θ( V) = X(θ(V))+ [α(X),θ(V)], (3.10)
X
∇ 2
where V Γ(ψ 1TG) and X X(M).
−
∈ ∈
(cid:3)
We show
6 HAJIME URAKAWA
Theorem 3.3. For every ψ C (M,G), we have
∞
∈
θ(τ (ψ)) = θ(J(τ(ψ)))
2
= δdδα Trace ([α,dδα]), (3.11)
g
− −
where α = ψ θ.
∗
Here, let us recall the definition:
Definition 3.4. For two g-valued 1-forms α and β on M, we define
g-valued symmetric 2-tensor [α,β] on M by
1
[α,β](X,Y) := [α(X),β(Y)] [β(X),α(Y)] , (X,Y X(M))
2 { − } ∈
(3.12)
and its trace Trace ([α,β]) by
g
m
Trace ([α,β]) := [α,β](e ,e ). (3.13)
g i i
i=1
X
We also use a g-valued 2-form [α β] on M by
∧
1
[α β](X,Y) := [α(X),β(Y)] [α(Y),β(X)] , (X,Y X(M)).
∧ 2 { − } ∈
(3.14)
Then, we have imediately by Theorem 3.3,
Corollary 3.5. For every ψ C (M,G), we have
∞
∈
(1) ψ : (M,g) (G,h) is harmonic if and only if
→
δα = 0. (3.15)
(2) ψ : (M,g) (G,h) is biharmonic if and only if
→
δdδα+Trace ([α,dδα]) = 0. (3.16)
g
We give a proof of Theorem 3.3.
Proof. (The first step) We first show that, for all V Γ(ψ 1TG),
−
∈
θ(∆V) = ∆ θ(V)
g
m 1
[e (α(e )),θ(V)]+[α(e ),e (θ(V))]
i i i i
− 2
i=1(cid:26)
X
1
+ [α(e ),[α(e ),θ(V)]]
i i
4
1
[α( e ),θ(V)] , (3.17)
−2 ∇ei i
(cid:27)
BIHARMONIC MAPS INTO COMPACT LIE GROUPS 7
where e m is a locally defined orthonormal frame field on (M,g),
{ i}i=1
and ∆ is the (positive) Laplacian of (M,g) acting on C (M).
g ∞
Indeed, we have by using Lemma 3.2 twice,
m
θ(∆V) = θ( ( V)) θ( V)
− ∇ei ∇ei − ∇∇eiei
Xi=1n o
m 1
= e (θ( V))+ [α(e ),θ( V)]
− i ∇ei 2 i ∇ei
i=1(cid:26)
X
1
e (θ(V)) [α( e ),θ(V)]
−∇ei i − 2 ∇ei i
(cid:27)
m 1
= e e (θ(V))+ [α(e ),θ(V)]
i i i
− 2
i=1(cid:26) (cid:18) (cid:19)
X
1 1
+ α(e ),e (θ(V))+ [α(e ),θ(V)]
i i i
2 2
(cid:20) (cid:21)
1
e (θ(V)) [α( e ),θ(V)]
−∇ei i − 2 ∇ei i
(cid:27)
m
= e (e (θ(V))) e (θ(V))
− { i i −∇ei i }
i=1
X
m 1 1
e ([α(e ),θ(V)])+ [α(e ),e (θ(V))]
i i i i
− 2 2
i=1(cid:26)
X
1
+ [α(e ),[α(e ),θ(V)]]
i i
4
1
[α( e ),θ(V)] . (3.18)
−2 ∇ei i
(cid:27)
Here, we have
e ([α(e ),θ(V)]) = [e (α(e )),θ(V)]+[α(e ),e (θ(V))],
i i i i i i
which wesubstitute into(3.18), andbydefinition of∆ , we have (3.17).
g
(The second step) On the other hand, we have to consider
m m
Rh(V,ψ e ))ψ e = Rh(L 1 V,L 1 ψ e )L 1 ψ e .
−i=1 ∗ i ∗ i −i=1 −ψ(x)∗ −ψ(x)∗ ∗ i −ψ(x)∗ ∗(3i.19)
X X
Under the identification T G Z Z g, we have
e e
∋ ↔ ∈
T G L 1 ψ e α(e ) g, (3.20)
e ∋ −ψ(x)∗ ∗ i ↔ i ∈
T G L 1 V θ(V) g, (3.21)
e ∋ −ψ(x) ↔ ∈
∗
respectively.
8 HAJIME URAKAWA
Because, we have
n
L 1 ψ e = h(ψ e ,X )X
−ψ(x) i i sψ(x) se
∗ ∗ i=1 ∗
X
and
α(e ) = ψ θ(e )
i ∗ i
= θ(ψ e )
i
∗
n
= h(ψ e ,X )θ(X )
i sψ(x) sψ(x)
∗
s=1
X
n
= h(ψ e ,X )X ,
i sψ(x) s
∗
s=1
X
which implies that (3.20). By the same way, we have (3.21).
Under this identification, the curvature tensor (G,h) is given as (see
Kobayashi-Nomizu ([5], pp. 203–204)),
1
Rh(X,Y) = ad([X,Y]) (X,Y g),
e
−4 ∈
and then, we have
m 1 m
θ Rh(V,ψ e ))ψ e = [[θ(V),α(e )],α(e )]
i i i i
− ∗ ∗ ! 4
i=1 i=1
X X
1 m
= [α(e ),[α(e ),θ(V)]].
i i
4
i=1 (3.22)
X
BIHARMONIC MAPS INTO COMPACT LIE GROUPS 9
(The third step) By (3.17) and (3.22), for V Γ(ψ 1TG), we have
−
∈
m
θ ∆V Rh(V,ψ e )ψ e
i i
− ∗ ∗ !
i=1
X
= ∆ θ(V)
g
m 1
[e (α(e )),θ(V)]+[α(e ),e (θ(V))]
i i i i
− 2
i=1(cid:26)
X
1
+ [α(e ),[α(e ),θ(V)]]
i i
4
1
[α( e ),θ(V)]
−2 ∇ei i
(cid:27)
1 m
+ [α(e ),[α(e ),θ(V)]]
i i
4
i=1
X
1 m m
= ∆ θ(V) e (α(e )),θ(V)]+ [α(e ),e (θ(V))]
g i i i i
− 2
i=1 i=1
X X
1 m
+ [α( e ),θ(V)]
2 ∇ei i
i=1
X
1 m
= ∆ θ(V) (e (α(e )) α( e )),θ(V)
g − 2 " i i − ∇ei i #
i=1
X
m
+ [α(e ),e (θ(V))]
i i
i=1
X
1 m
= ∆ θ(V)+ [δα,θ(V)]+ [α(e ),e (θ(V))]. (3.23)
g i i
2
i=1
X
(The forth step) For V = τ(ψ) in (3.23), since θ(τ(ψ)) = δα, we
−
have
1
θ(J(τ(ψ))) = ∆ (θ(τ(ψ))+ [δα,θ(τ(ψ))]
g
2
m
+ [α(e ),e (θ(τ(ψ))]
i i
i=1
X
1 m
= ∆ δα [δα,δα] [α(e ),e (δα)]
g i i
− − 2 −
i=1
X
m
= ∆ δα [α(e ),e (δα)]
g i i
− −
i=1
X
m
= ∆ δα [α(e ),dδα(e )]. (3.24)
g i i
−
i=1
X
10 HAJIME URAKAWA
(cid:3)
Then, (3.24) implies the desired (3.11).
4. Biharmonic maps from R into compact Lie groups
Inthissection,weconsiderthesimplestcase: thestandard1-dimensional
Euclidean space, (M,g) = (R,g ) and (G,h), an n-dimensional com-
0
pact Lie group with the bi-invariant Riemannian metric h.
4.1 First, let ψ : R t ψ(t) (G,h), a C curve in G. Then,
∞
α := ψ θ is a g-valued 1-∋form7→on R.∈So, α can be written at t R as
∗
∈
α = F(t)dt (4.1)
t
where F : R t F(t) g is a C function on R given by
∞
∋ 7→ ∈
∂ ∂ ∂
F(t) = α = ψ θ = θ ψ . (4.2)
∗
∂t! ∂t! ∗ ∂t!!
Here, since
∂ n ∂
ψ (t) := ψ = h ψ ,X X ,
′ ∗ ∂t! i=1 ψ(t) ∗ ∂t! sψ(t)! sψ(t) (4.3)
X
we have
n ∂
F(t) = h ψ ,X X , (4.4)
ψ(t) ∗ ∂t! sψ(t)! s
s=1
X
so that we have the following correspondence:
n
T G L 1 ψ (t) = h (ψ (t),X )X
e ∋ −ψ(t) ′ ψ(t) ′ sψ(t) se
∗ s=1
X
∂
F(t) = θ ψ g. (4.5)
↔ ∗ ∂t!! ∈
4.2 We first show that
δα = F (t). (4.6)
′
−
Proof. Indeed, we have
δα = h(α(e )) α( e )
−{∇ei i − ∇e1 1 }
= e (α(e ))
1 1
−
= e (F(t))
1
−
= F (t).
′
−
(cid:3)