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BIHARMONIC MAPS INTO SYMMETRIC SPACES AND INTEGRABLE SYSTEMS 2 1 HAJIME URAKAWA 0 2 n a J Abstract. Inthispaper,thedescriptionofbiharmonicmapequa- tion in terms of the Maurer-Cartan form for all smooth map of a 1 3 compactRiemannianmanifoldintoaRiemanniansymmetricspace (G/K,h)inducedfromthebi-invariantRiemannianmetrichonG ] isobtained. Bythisformula,allbiharmoniccurvesintosymmetric G spaces are determined, and all the biharmonic maps of an open D domain ofR2 with the standardRiemannianmetric into (G/K,h) h. are characterized. t a m 1. Introduction and statement of results. [ This paper is a continuation of our previous one [15]. In our previous 2 paper, we discussed about the description of biharmonic maps into v 2 compact Lie groups in terms of the Maurer-Cartan form, and gave 5 their explicit constructions. In this paper, we want to extend them to 1 3 biharmonic maps into Riemannian symmetric spaces. . The theory of harmonic maps into Lie groups, symmetric spaces or 1 0 homogeneous spaces has been extensively studied related to the in- 1 tegrable systems by many authors (for instance, [14], [17], [3], [18], 1 : [9], [10], [11], [13], [2]). In particular, the moduli space of harmonic v maps of 2-sphere into symmetric spaces was completely determined i X (cf. [3], [5], [13]). Let us recall the loop group formulation of harmonic r a maps into symmetric spaces, briefly. Let ϕ be a smooth map of a Rie- mann surface M into a Riemannian symmetric space (G/K,h) with a lift ψ : M G so that π ψ = ϕ. Let g = k m be the corre- → ◦ ⊕ sponding Cartan decomposition of the Lie algebra g of the Lie group G. Then, the pull back α = ψ 1dψ of the Maurel-Cartan form on G − is decomposed as α = α + α , correspondingly. Let us decompose k m α into the sum of holomorphic part and the anti-holomorphic one: m 2000 Mathematics Subject Classification. 58E20. Key words and phrases. harmonic map, biharmonic map, symmetric space, in- tegrable system, Maurer-Cartanform. Supported by the Grant-in-Aid for the Scientific Research, (C), No. 21540207, Japan Society for the Promotionof Science. 1 2 HAJIME URAKAWA α = α +α . Then, one can obtain the extended solution ψ of M m m′ m′′ intoaloopgroupΛGsatisfying that ψ 1dψ = λα +α +λ 1α forall − m′ k − m′′ λ U(1) = λ C : λ = 1 . (cf. [3]). Then, ϕ : M (G/Ke,h) is ∈ { ∈ | | } e e → harmonicifandonlyifthereexistsaholomorphicandhorizontalmapψ of M into the homogeneous ΛG/K with ψ = ψ (cf. [3], p. 648). Then, 1 e one can obtain Weierstrass-type representation of harmonic maps (cf. e [3], pp. 648–662). On the other hand, the notion of harmonic map has been extended to the one of biharmonic map (cf. [4], [7]). In this paper, we wil describe biharmonic maps into Riemannian symmetric spaces in terms of the pull back α = α +α of the Maurer-Cartan form (cf. Theorem k m 3.6), give some explicit solutions of the biharmonic map equation in Riemannian symmetric spaces, and construct several biharmonic maps into Riemannian symmetric spaces (Sections 4 and 5). Acknowledgement: The author expresses his gratitude to Prof. J. Inoguchi and Prof. Y. Ohnita who gave many useful suggestions and discussions, and Prof. A. Kasue for his financial support during the preparation of this paper. 2. Preliminaries. In this section, we prepare general materials and facts harmonic maps, biharmonic maps into Riemannian symmetric spaces (cf. [8]). 2.1. Let (M,g) be an m-dimensional compact Riemannian manifold, and the target space (N,h) is an n-dimensional Riemannian symmet- ric space (G/K,h). Nemely, let g, k be the Lie algebras of G, K, and g = k m is the Cartan decomposition of g, and h, the G-invariant ⊕ Riemannian metric on G/K corresponding to the Ad(K)-invariant in- ner product , on m. Let k be a left invariant Riemannian metric h i on G such as the natural projection π : G G/K is a Riemannian → submersion of (G,k) onto (G/K,h). For every C map ϕ of M into ∞ G/K, let us take its (local) lift ψ : M G of ϕ, i.e., ϕ = π ψ, → ◦ ϕ(x) = ψ(x)K G/K (x U M), where U is an open subset of ∈ ∈ ⊂ M. The energy functional on the space C (M,G/K) of all C maps of ∞ ∞ M into G/K is defined by 1 E(ϕ) = dϕ 2v , g 2 M | | Z BIHARMONIC MAPS INTO SYMMETRIC SPACES 3 and for a C one parameter deformation ϕ C (M,G/K) ( ǫ < ∞ t ∞ ∈ − t < ǫ) of ϕ with ϕ = ϕ, the first variation formula is given by 0 d E(ϕ ) = τ(ϕ),V v , t g dt(cid:12) − Mh i (cid:12)t=0 Z (cid:12) where V is a variatio(cid:12)n vector field along ϕ defined by V = d ϕ (cid:12) dt t=0 t which belongs to the space Γ(ϕ 1T(G/K)) of sections of the ind(cid:12)uced − (cid:12) bundle of the tangent bundle T(G/K) by ϕ. The tension field τ(cid:12)(ϕ) is defined by m τ(ϕ) = B(ϕ)(e ,e ), (2.1) i i i=1 X where B(ϕ)(X,Y) = h dϕ(Y) dϕ( Y) ∇dϕ(X) − ∇X for X,Y X(M). Here, , and h, are the Levi-Civita connections of ∈ ∇ ∇ (M,g) and (G/K,h), respectively. For a harmonic map ϕ : (M,g) → (G/K,h), the second variation formula of the energy functional E(ϕ) is d2 E(ϕ ) = J(V),V v t g dt2(cid:12) Mh i (cid:12)t=0 Z where (cid:12) (cid:12) (cid:12) J(V) := ∆V (V), (2.2) −R m ∆V := ∗ V = ( V) V , (2.3) ∇ ∇ − {∇ei ∇ei −∇∇eiei } i=1 X m (V) := Rh(V,dϕ(e ))dϕ(e ). (2.4) i i R i=1 X Here, is the induced connection on the induced bundle ϕ 1T(G/K), − ∇ and is Rh is the curvature tensor of (G/K,h) given by Rh(U,V)W = [ h, h]W h W (U,V,W X(G/K)). ∇U ∇V −∇[U,V] ∈ The bienergy functional is defined by 1 1 E (ϕ) = (d+δ)2ϕ 2v = τ(ϕ) 2v , (2.5) 2 g g 2 M | | 2 M | | Z Z and the first variation formula of the bienergy is given (cf. [7]) by d E (ϕ ) = τ (ϕ),V v (2.6) 2 t 2 g dt(cid:12) − Mh i (cid:12)t=0 Z (cid:12) where the bitension fi(cid:12)eld τ (ϕ) is defined by (cid:12) 2 τ (ϕ) = J(τ(ϕ)) = ∆τ(ϕ) (τ(ϕ)), (2.7) 2 −R 4 HAJIME URAKAWA and a C map ϕ : (M,g) (G/K,h) is called to be biharmonic if ∞ → τ (ϕ) = 0. (2.8) 2 2.2. Let k be a left invariant Riemannian metric on G corresponding to the inner product , on g given by , = B( , ) if (G/K,h) h· ·i h· ·i − · · is of compact type, and by U + X,V + Y = B(U,V) + B(X,Y) h i − (U,V k, X,Y m) if (G/K,h) is of non-compact type. Here, B( , ) ∈ ∈ · · is the Killing form of g. Then, the projection π of G onto G/K is a Riemannian submersion of (G,k) onto (G/K,h), and we have also the orthogonal decomposition of the tangent space T G (x M) with ψ(x) ∈ respect to the inner product k ( , ) (x M) in such a way that ψ(x) · · ∈ T G = V H , (2.9) ψ(x) ψ(x) ψ(x) ⊕ where the vertical space at ψ(x) G is given by ∈ V = Ker(π ) = X X k , (2.10) ψ(x) ψ(x) ψ(x) ∗ { | ∈ } and the horizontal space at ψ(x) is given by H = Y Y m , (2.11) ψ(x) ψ(x) { | ∈ } corresponding to the Cartan decomposition g = k m. Then, for every ⊕ C section W Γ(ψ 1TG), we have the decomposition corresponding ∞ − ∈ to (2.9), W(x) = WV(x)+WH(x) (x M), (2.12) ∈ where WV, WH, (denoted also by W, W, respectively) belong to V H Γ(ψ 1TG). Wedenote byΓ(E), the spaceof allC sections ofavector − ∞ bundle E. For Y m, define Y Γ(ψ 1TG) by Y(x) := Y (x − ψ(x) ∈ ∈ ∈ M). Let X n be an orthonormal basis of m with respect to the inner prod{ucit}i=,1 of g correspeonding to the left inevariant Riemannian h· ·i metric k on G. Then, WH can be written in terms of X as i n WH = f X f i i i=1 X wheref C (M)(i = 1, ,n). Becaufse, foreveryx M, WH(x) i ∞ ∈ ··· ∈ ∈ H , so that we have ψ(x) n n WH(x) = f (x)X = f (x)X (x). i iψ(x) i i i=1 i=1 X X We say W Γ(ψ 1TG) and V Γ(ϕ 1T(Gf/K)) are π-related, − − ∈ ∈ denoted by V = π W, if it holds that ∗ V(x) = π W(x) (x M), ∗ ∈ BIHARMONIC MAPS INTO SYMMETRIC SPACES 5 whereπ : T G T (G/K) = T (G/K)isthedifferentiation ψ(x) ϕ(x) π(ψ(x)) ∗ → of the projection π of G onto G/K at ψ(x) for each x M. ∈ Let be , k, h, the Levi-Civita connections of (M,g), (G,k), ∇ ∇ ∇ (G/K,h), and , , the induced connection of k on the induced ∇ ∇ ∇ bundle ψ 1TG by ψ : M G, and the one of h on the induced − → ∇ bundle ϕ 1T(G/K) by ϕ : M G/K, respectively. − → Lemma 2.1. Assume that W Γ(ψ 1TG) and V Γ(ϕ 1T(G/K)) − − ∈ ∈ are π-related, i.e., V = π W. ∗ (1) Then, we have V = π k WH, (2.13) ∇X ∗∇(ψ∗X)H where (ψ X)H is the horizontal component of ψ X for every C vector ∞ ∗ ∗ field X on M. (2) If we express WH = n f X and (ψ X)H = n g X where i=1 i i j=1 j j ∗ f , g C (M)(i,j = 1, ,n), then, it holds that i j ∈ ∞ ··P· f P f 1 n n k WH = f (x)g (x)[X ,X ] + X (f )X (x) ∇(ψ∗X)H ψ(x) 2 i j j i ψ(x) x i i (cid:16) (cid:17) iX,j=1 Xi=1 f V H (x M), (2.14) ψ(x) ψ(x) ∈ ⊕ ∈ correspondingly. (3) For every x M, we have ∈ n V(x) = X ( W,X )π (X ). (2.15) X x iψ(x) iψ(x) ∇ h i ∗ i=1 X Here, it holds that π (X ) = t π (X) (X m), where t is the ψ(x) ψ(x) a ∗ ∗ ∗ ∈ translation of G/K by a G, i.e., t (yK) := ayK (y G). a ∈ ∈ Proof. (1) Due to Lemmas 1 and 3 in [13], p.460, we have V = h V ∇X ∇ϕ∗X = h π W ∇π∗(ψ∗X) ∗ = π k WH ∗ H∇(ψ∗X)H = π (cid:16)k WH. (cid:17) ∗∇(ψ∗X)H 6 HAJIME URAKAWA (2) Indeed, we have n k WH = g (x) k WH ∇(ψ∗X)H ψ(x) j ∇Xj ψ(x) (cid:16) (cid:17) jX=1 (cid:16) (cid:17) n n = g (x) k (f X ) j ∇Xj i i ! jX=1 Xi=1 ψ(x) n f = g (x) (X f )(x)X (x)+f (x) k X j j i i i ∇Xj i ψ(x) i,Xj=1 (cid:26) (cid:16) (cid:17) (cid:27) f 1 = g (x) (X f )(x)X (x)+ f (x)[X ,X ] j j i i i j i ψ(x) 2 i,j=1 (cid:26) (cid:27) X 1 n f n = f (x)g (x)[X ,X ] + X (f )X (x), i j j i ψ(x) x i i 2 i,j=1 i=1 X X f since it holds that k X = L k X ∇Xj i ψ(x) ψ(x)∗ ∇Xj i e (cid:16) (cid:17) (cid:16) (cid:17) 1 = L [X ,X ] ψ(x) j i e ∗ 2 (cid:18) (cid:19) 1 = [X ,X ] j i ψ(x) 2 and [m,m] k. For (3), notice that WH = n W,X X . Due to ⊂ i=1h iψ(·)i i (cid:3) (1), (2), we have (3). P f Lemma 2.2. Under the same assumption of Lemma 2.1, we have, n ( V) = X (Y W,X )π (X ) T (G/K), X Y x iψ() iψ(x) ϕ(x) ∇ ∇ h · i ∗ ∈ i=1 (2.16) X at each x M, for every C vector fields X and Y on M. ∞ ∈ Proof. Let Z := V Γ(ϕ 1T(G/K)). Then, by Lemma 2.1 (1), we Y − ∇ ∈ have ( V) = Z = π k ZH (2.17) ∇X ∇Y ∇X ∗∇(ψ∗X)H where by Lemma 2.1 (3), we have for every y M, ∈ n ZH(y) = Y W,X X , y iψ() iψ(y) h · i i=1 X Z(y) = π ZH(y) T (G/K) and Z Γ(ϕ 1T(G/K)). Then, at ϕ(y) − ∗ ∈ ∈ each x M, the right hand side of (2.17) which belong to T (G/K), ϕ(x) ∈ BIHARMONIC MAPS INTO SYMMETRIC SPACES 7 coincides with the following: n X ZH,X π (X ) x jψ() jψ(x) h · i ∗ j=1 X n n = X Y W,X X ,X π (X ) x iψ iψ() jψ() jψ(x) h •h i · · i ∗ j=1 i=1 X X n = X (Y W,X ) δ π (X ) x iψ ij jψ(x) •h i ∗ i,j=1 X n = X (Y W,X ) π (X ). x iψ jψ(x) •h i ∗ i=1 X (cid:3) Thus, we have (2.16). Proposition 2.3. The rough Laplacian ∆ acting on Γ(ϕ 1T(G/K)) − can be calculated as follows: For V Γ(ϕ 1T(G/K)) with V = π W − for W Γ(ψ 1TG), ∈ ∗ − ∈ n (∆V)(x) = ∆ W,X π (X ) T (G/K), x iψ() iψ(x) ϕ(x) h · i ∗ ∈ i=1 (2.18) X for each x M. Here, since f : M x W(x),X R is a iψ(x) ψ(x) ∈ ∋ 7→ h i ∈ (local) C function on M, the Laplacian ∆ = δd acting on C (M) ∞ x ∞ works on f. Indeed, if we recall the definition (2.3) of the rough Laplacian ∆, and due to Lemmas 2.1 and 2.2, we have m ∆V = ( V) V , − {∇ej ∇ej −∇∇ejej } j=1 X n n = (e 2 e ) W,X π (X ) − j −∇ej j h iψ(·)i ∗ iψ(x) i=1j=1 XX n = ∆ W,X π (X ). x iψ() iψ(x) h · i ∗ i=1 X (cid:3) We have Proposition 2.3. 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 ϕ of (M,g) into (G/K,h) with a lift ψ : M G, ∞ → 8 HAJIME URAKAWA let us consider a g-valued 1-form α on M given by α = ψ θ and the ∗ decomposition α = α +α (3.1) k m corresponding to the decomposition g = k m. Then, it is well known ⊕ (see for example, [2]) that Lemma 3.1. For every C map ϕ : (M,g) (G/K,h), ∞ → m tψ(x)−1 τ(ϕ) = δ(αmj + [αk(ei),αm(ei)], (x M), ∗ − ∈ i=1 (3.2) X where α = ϕ θ, and θ is the Maurer-Cartan form of G, δ(α ) is the ∗ m co-differentiation of m-valued 1-form α on (M,g). m Thus, ϕ : (M,g) (G/K,h) is harmonic if and only if → m δ(α )+ [α (e ),α (e )] = 0. (3.3) m k i m i − i=1 X Furthermore, we obtain Theorem 3.2. We have m tψ(x)−1 τ2(ϕ) = ∆g δ(αm)+ [αk(ei),αm(ei)] ∗ − ! i=1 X m m + δ(α )+ [α (e ),α (e )],α (e ) ,α (e ) , m k i m i m s m s s=1""− i=1 # # (3.4) X X where ∆ is the (positive) Laplacian of (M,g) acting on C functions g ∞ on M, and e m is a local orthonormal frame field on (M,g). { i}i=1 Therefore, we obtain immediately the following two corollaries. Corollary 3.3. Let (G/K,h) be a Riemannian symmetric space, and ϕ : (M,g) (G/K,h), a C mapping. Then, we have: ∞ → (1) the map ϕ : (M,g) (G/K,h) is harmonic if and only if → m δ(α )+ [α (e ),α (e )] = 0. (3.5) m k i m i − i=1 X BIHARMONIC MAPS INTO SYMMETRIC SPACES 9 (2) The map ϕ : (M,g) (G/K,h) is biharmonic if and only if → m ∆ δ(α )+ [α (e ),α (e )] g m k i m i − ! i=1 X m m + δ(α )+ [α (e ),α (e )],α (e ) ,α (e ) = 0. m k i m i m s m s s=1""− i=1 # # (3.6) X X Corollary 3.4. Let (G/K,h) be a Riemannian symmetric space, and ϕ : (M,g) (G/K,h), a C mapping with a horizontal lift ψ : M ∞ → → G, i.e., ϕ = π ψ and ψ (T M) H which is equivalent to α 0. x x ψ(x) k ◦ ⊂ ≡ Then, we have: (1) the map ϕ : (M,g) (G/K,h) is harmonic if and only if → δ(α ) = 0, (3.7) m (2) and the map ϕ : (M,g) (G/K,h) is biharmonic if and only if → m δdδ(α )+ [[δ(α ),α (e )],α (e )] = 0. (3.8) m m m s m s s=1 X Proof of Theorem 3.2. We need the following lemma: Lemma 3.5. The tension field τ(ϕ) of a C map ϕ : (M,g) ∞ → (G/K,h) can be expressed as τ(ϕ) = π W = π (WH), ∗ ∗ where W Γ(ψ 1TG), and WH is the horizontal component of W in − ∈ the decomposition W(x) = WV(x)+WH(x) T G = V H ψ(x) ψ(x) ψ(x) ∈ ⊕ (x M). If we define an m-valued function β on M by ∈ n n β := W,X X = WH,X X , (3.9) i i i i h i h i i=1 i=1 X X then, we have f f t 1τ(ϕ) = π β. (3.10) ψ(x) − ∗ ∗ If we define n m-valued functions β (i = 1, ,n) on M by i ··· n β := ψ e ,X X m. (3.11) i i jψ() j h ∗ · i ∈ j=1 X Then, it holds that t 1ϕ e = π β and β = α (e ), (3.12) ψ(x) − i i i m i ∗ ∗ ∗ 10 HAJIME URAKAWA where α is the m-component of α := ψ θ, and the Maurer-Cartan m ∗ form on G. Indeed, (3.10) and the first part of (3.12) follow from the definition of β and the fact that α(e ) = (ψ θ)(e ) i ∗ i = θ(ψ e ) i ∗ n ℓ = θ ψ e ,X X + ψ e ,X X i jψ(x) jψ(x) i jψ(x) jψ(x)  h ∗ i h ∗ i  j=1 j=n+1 X X n ℓ  = ψ e ,X X + ψ e ,X X i jψ(x) j i jψ(x) j h ∗ i h ∗ i j=1 j=n+1 X X m k, ∈ ⊕ since α = ψ θ. Thus, we have β = α (e ). (cid:3) ∗ i m i (Continued the proof of Theorem 3.2) We have n t 1ϕ e = ψ e ,X π (X ) T (G/K), ψ(x) − i i jψ(x) j o ∗ ∗ h ∗ i ∗ ∈ j=1 (3.13) X where o = K G/K is the origin of G/K. Because, { } ∈ t 1ϕ e = t 1π ψ e = π L 1ψ e = π (L 1ψ e ) ψ(x) − i ψ(x) − i ψ(x) − i ψ(x) − i m ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ which coincides with n n L 1ψ e ,X π (X ) = ψ e ,X π (X ), ψ(x) − i j j i jψ(x) j h ∗ ∗ i ∗ h ∗ i ∗ j=1 j=1 X X which imply (3.13). Thus, we have ϕ e = π W (i = 1, ,m), (3.14) i i ∗ ∗ ··· whereW Γ(ψ 1TG)andm-valuedfunctionsW onM (i = 1, ,m) i − i ∈ ··· are given by f n W (x) := ψ e ,X X , (3.15) i i jψ(x) jψ(x) h ∗ i j=1 X n W (x) := ψ e ,X X m, (3.16) i i jψ(x) j h ∗ i ∈ j=1 X g for each x M. ∈

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