Harmonic spheres in outer symmetric spaces, their canonical elements and Weierstrass type representations
aa r X i v : . [ m a t h . DG ] M a r HARMONIC SPHERES IN OUTER SYMMETRIC SPACES, THEIR CANONICALELEMENTS AND WEIERSTRASS-TYPE REPRESENTATIONS
N. CORREIA AND R. PACHECO
Abstract.
Making use of Murakami’s classification of outer involutions in a Lie algebra and following theMorse-theoretic approach to harmonic two-spheres in Lie groups introduced by Burstall and Guest, we obtaina new classification of harmonic two-spheres in outer symmetric spaces and a Weierstrass-type representationfor such maps. Several examples of harmonic maps into classical outer symmetric spaces are given in terms ofmeromorphic functions on S . Introduction
The harmonicity of maps ϕ from a Riemann surface M into a compact Lie group G with identity e amountsto the flatness of one-parameter families of connections. This establishes a correspondence between such mapsand certain holomorphic maps Φ into the based loop group Ω G , the extended solutions [17]. Evaluating anextended solution Φ at λ = − ϕ into the Lie group. If an extended solution takesvalues in the group of algebraic loops Ω alg G , the corresponding harmonic map is said to have finite unitonnumber . It is well known that all harmonic maps from the two-sphere into a compact Lie group have finiteuniton number [17].Burstall and Guest [1] have used a method suggested by Morse theory in order to describe harmonic mapswith finite uniton number from M into a compact Lie group G with trivial centre. One of the main ingredientsin that paper is the Bruhat decomposition of the group of algebraic loops Ω alg G . Each piece U ξ of the Bruhatdecomposition corresponds to an element ξ in the integer lattice I ( G ) = (2 π ) − exp − ( e ) ∩ t and can bedescribed as the unstable manifold of the energy flow on the K¨ahler manifold Ω alg G . Each extended solutionΦ : M → Ω alg G takes values, off some discrete subset D of M , in one of these unstable manifolds U ξ and can bedeformed, under the gradient flow of the energy, to an extended solution with values in some conjugacy class ofa Lie group homomorphism γ ξ : S → G . A normalization procedure allows us to choose ξ among the canonicalelements of I ( G ); there are precisely 2 n canonical elements, where n = rank( G ), and consequently 2 n classesof harmonic maps. Burstall and Guest [1] introduced also a Weierstrass-type representation for such harmonicmaps in terms of meromorphic functions on M . It is possible to define a similar notion of canonical element forcompact Lie groups G with non-trivial centre [5, 6]. In the present paper, we will not assume any restrictionon the centre of G .Given an involution σ of G , the compact symmetric G -space N = G/G σ , where G σ is the subgroup of G fixed by σ , can be embedded totally geodesically in G via the corresponding Cartan embedding ι σ . Henceharmonic maps into compact symmetric spaces can be interpreted as special harmonic maps into Lie groups.For inner involutions σ = Ad( s ), where s ∈ G is the geodesic reflection at some base point x ∈ N , thecomposition of the Cartan embedding with left multiplication by s gives a totally geodesic embedding of G/G σ in G as a connected component of √ e . Reciprocally, any connected component of √ e is a compact innersymmetric G -space. As shown by Burstall and Guest [1], any harmonic map into a connected component of √ e admits an extended solution Φ which is invariant under the involution I (Φ)( λ ) = Φ( − λ )Φ( − − . Offa discrete set, Φ takes values in some unstable manifold U ξ and can be deformed, under the gradient flowof the energy, to an extended solution with values in some conjugacy class of a Lie group homomorphism γ ξ : S → G σ . An appropriate normalization procedure which preserves both I -invariance and the underlying connected component of √ e allows us to choose ξ among the canonical elements of I ( G σ ). As a matter of fact,since σ is inner, rank( G ) = rank( G σ ) and we have I ( G ) = I ( G σ ), that is the canonical elements of I ( G ) coincidewith the canonical elements of I ( G σ ). Consequently, if G has trivial center, we have 2 n classes of harmonicmaps with finite uniton number into all inner symmetric G -spaces.The theory of Burstall and Guest [1] on harmonic two-spheres in compact inner symmetric G -spaces wasextended by Eschenburg, Mare and Quast [8] to outer symmetric spaces as follows: each harmonic map from atwo-sphere into an outer symmetric space G/G σ , with outer involution σ , corresponds to an extended solutionΦ which is invariant under a certain involution T σ induced by σ on Ω G (see also [11]); Φ takes values in someunstable manifold U ξ , off some discrete set; under the gradient flow of the energy any such invariant extendedsolution is deformed to an extended solution with values in some conjugacy class of a Lie group homomorphism γ ξ : S → G σ ; applying the normalization procedure of extended solutions introduced by Burstall and Guestfor Lie groups, ξ can be chosen among the canonical elements of I ( G σ ) ( I ( G ); if G has trivial centre, thereare precisely 2 k canonical homorphisms, where k = rank( G σ ) < rank( G ); hence there are at most k classes ofharmonic two-spheres in G/G σ if G has trivial centre.In the present paper, we will establish a more accurate classification of harmonic maps from a two-sphereinto compact outer symmetric spaces. This classification takes into consideration the following crucial factsconcerning extended solutions associated to harmonic maps into outer symmetric spaces: although any harmonicmap from a two-sphere into an outer symmetric space G/G σ admits a T σ -invariant extended solution, not all T σ -invariant extended solutions correspond to harmonic maps into G/G σ ; on the other hand, the Burstall andGuest’s normalization procedure does not necessarily preserve T σ -invariance.Our strategy is the following. The existence of outer involutions of a simple Lie algebra g depends on theexistence of non-trivial involutions of the Dynkin diagram of g C [2, 8, 12, 14]. More precisely, if ̺ is a non-trivial involution of the Dynkin diagram of g C , then it induces an outer involution σ ̺ of g C , which we callthe fundamental outer involution , and, as shown by Murakami [14], all the other outer involutions are, up toconjugation, of the form σ ̺,i := Ad exp πζ i ◦ σ ̺ where each ζ i is a certain element in the integer lattice I ( G σ ̺ ).Each connected component of P σ ̺ = { g ∈ G | σ ̺ ( g ) = g − } is a compact outer symmetric G -space associatedto some involution σ ̺ or σ ̺,i ; reciprocally, any outer symmetric space G/G σ , with σ equal to σ ̺ or σ ̺,i , canbe totally geodesically embedded in the Lie group G as a connected component of P σ ̺ (see Proposition 6).As shown in Section 4.2, any harmonic map ϕ into a connected component N of P σ ̺ admits a T σ ̺ -invariantextended solution Φ; off a discrete set, Φ takes values in some unstable manifold U ξ . In Section 4.2.2 weintroduce an appropriate normalization procedure in order to obtain from Φ a normalized extended solution ˜Φwith values in some unstable manifold U ζ such that: ζ is a canonical element of I ( G σ ̺ ); ˜Φ is T τ -invariant, where τ is the outer involution given by τ = Ad exp π ( ξ − ζ ) ◦ σ ̺ ; ˜Φ( −
1) takes values in some connected componentof P σ ̺ which is an isometric copy of N completely determined by ζ and τ ; moreover, ˜Φ( −
1) coincides with ϕ up to isometry. Hence, we obtain a classification of harmonic maps of finite uniton number from M into outersymmetric G -spaces in terms of the pairs ( ζ, τ ).Dorfmeister, Pedit and Wu [7] have introduced a general scheme for constructing harmonic maps from aRiemann surface into a compact symmetric space from holomorphic data, in which the harmonic map equationreduces to a linear ODE similar to the classical Weierstrass representation of minimal surfaces. Burstal andGuest [1] made this scheme more explicit for the case M = S by establishing a “Weierstrass formula” forharmonic maps with finite uniton number into Lie groups and their inner symmetric spaces. In Theorem 17we establish a version of this formula to outer symmetric spaces, which allows us to describe the corresponding T σ -invariant extended solutions in terms of meromorphic functions on M . For normalized extended solutionsand “low uniton number”, such descriptions are easier to obtain. In Section 5 we give several explicit examplesof harmonic maps from the two-sphere into classical outer symmetric spaces: Theorem 20 interprets old resultsby Calabi [3] and Eells and Wood [9] concerning harmonic spheres in real projective spaces R P n − in viewof our classification; harmonic two-spheres into the real Grassmannian G ( R ) are studied in detail; we showthat all harmonic two spheres into the Wu manifold SU (3) /SO (3) can be obtained explicitly by choosing two meromorphic functions on S and then performing a finite number of algebraic operations, in agreement withthe explicit constructions established by H. Ma in [13].2. Groups of algebraic loops
For completeness, in this section we recall some fundamental facts concerning the structure of the group ofalgebraic loops in a compact Lie group. Further details can be found in [1, 4, 15].Let G be a compact matrix semisimple Lie group with Lie algebra g and identity e . Denote the free and based loop groups of G by Λ G and Ω G , respectively, whereas Λ + G C stands for the subgroup of Λ G C consistingof loops γ : S → G C which extend holomorphically to the unitary disc | λ | < T of G with Lie algebra t ⊂ g . Let ∆ ⊂ i t ∗ be the corresponding set of roots, wherei := √−
1, and, for each α ∈ ∆, denote by g α the corresponding root space. Choose a fundamental Weyl chamber W in t , which corresponds to fix a positive root system ∆ + . The intersection I ′ ( G ) := I ( G ) ∩ W parameterizesthe conjugacy classes of homomorphisms S → G . More precisely, Hom( S , G ) is the disjoint union of Ω ξ ( G ),with ξ ∈ I ′ ( G ), where Ω ξ ( G ) is the conjugacy class of homomorphisms which contains γ ξ ( λ ) = exp ( − i ln( λ ) ξ ).The Bruhat decomposition states that the subgroup of algebraic based loops Ω alg G is the disjoint union ofthe orbits Λ +alg G C · γ ξ , with ξ ∈ I ′ ( G ), where · denotes the dressing action of Λ + G C on Ω G induced by the Iwasawa decomposition Λ G C ∼ = Ω G × Λ + G C . According to the Morse theoretic interpretation [1, 15] of theBruhat decomposition, for each ξ ∈ I ′ ( G ), U ξ ( G ) := Λ +alg G C · γ ξ is the unstable manifold of Ω ξ ( G ) under theflow induced by the energy gradient vector field −∇ E : each γ ∈ U ξ ( G ) flows to some homomorphism u ξ ( γ ) inΩ ξ ( G ).In [1], the authors proved that, for each ξ ∈ I ′ ( G ), the critical manifold Ω ξ ( G ) is a complex homogeneousspace of G C and the unstable manifold U ξ ( G ) is a complex homogeneous space of the group Λ +alg G C . Moreover, U ξ ( G ) carries a structure of holomorphic vector bundle over Ω ξ ( G ) and the bundle map u ξ : U ξ ( G ) → Ω ξ ( G ) isprecisely the natural projection given by [ γ ] [ γ (0)].Define a partial order (cid:22) over I ′ ( G ) as follows: ξ (cid:22) ξ ′ if p ξi ⊂ p ξ ′ i for all i ≥
0, where p ξi = P j ≤ i g ξj and g ξj is the j i-eigenspace of ad ξ . As shown in [4], one can define a Λ +alg G C -invariant fibre bundle morphism U ξ,ξ ′ : U ξ ( G ) → U ξ ′ ( G ) by U ξ,ξ ′ (Ψ · γ ξ ) = Ψ · γ ξ ′ , Ψ ∈ Λ +alg G C , whenever ξ (cid:22) ξ ′ . Since the holomorphic structures on U ξ ( G ) and U ξ ′ ( G ) are induced by the holomorphicstructure on Λ +alg G C , the fibre-bundle morphism U ξ,ξ ′ is holomorphic.3. Harmonic spheres in Lie groups
Harmonic maps from the two-sphere S into a compact matrix Lie group G can be classified in terms ofcertain pieces of the Bruhat decomposition of Ω alg G . Next we recall briefly this theory from [1, 4, 5, 6].3.1. Extended Solutions.
Let M be a simply-connected Riemann surface, ϕ : M → G be a smooth map and ρ : G → End( V ) a finite representation of G . Equip G with a bi-invariant metric. If ϕ is an harmonic map of finite uniton number , it admits an extended solution Φ : M → Ω G with Φ( M ) ⊆ Ω alg G and ϕ = Φ − . In thiscase, we can write ρ ◦ Φ = P si = r ζ i λ i for some r ≤ s ∈ Z . The number s − r is called the uniton number ofΦ with respect to ρ , and the minimal value of s − r (with respect to all extended solutions associated to ϕ ) iscalled the uniton number of ϕ with respect to ρ and it is denoted by r ρ ( ϕ ). As explained in [6], this definition ofuniton number of an extended solution with respect to the adjoint representation is twice that of Burstall andGuest [1]. K. Uhlenbeck [17] proved that all harmonic maps from the two-sphere have finite uniton number.For simplicity of exposition, henceforth we will take M = S . However, all our results still hold for harmonicmaps of finite uniton number from an arbitrary Riemann surface. Theorem 1. [1] Let Φ : S → Ω alg G be an extended solution. Then there exists some ξ ∈ I ′ ( G ), and somediscrete subset D of S , such that Φ( S \ D ) ⊆ U ξ ( G ). N. CORREIA AND R. PACHECO
Given a smooth map Φ : S \ D → U ξ ( G ), consider Ψ : S \ D → Λ +alg G C such that Φ = Ψ · γ ξ , that isΨ γ ξ = Φ b for some b : S \ D → Λ +alg G C . Write Ψ − Ψ z = P i ≥ X ′ i λ i , and Ψ − Ψ ¯ z = P i ≥ X ′′ i λ i . Proposition4.4 in [1] establishes that Φ is an extended solution if, and only if,Im X ′ i ⊂ p ξi +1 , Im X ′′ i ⊂ p ξi , (1)where p ξi = L j ≤ i g ξj and g ξj is the j i-eigenspace of ad ξ . The derivative of the harmonic map ϕ = Φ − is givenby the following formula. Lemma 2. [4] Let Φ = Ψ · γ ξ : S → Ω alg G be an extended solution and ϕ = Φ − : S → G the correspondingharmonic map. Then ϕ − ϕ z = − X i ≥ b (0) X ′ ii +1 b (0) − , where X ′ ii +1 is the component of X ′ i over g ξi +1 , with respect to the decomposition g C = L g ξj .Both the fiber bundle morphism U ξ,ξ ′ : U ξ ( G ) → U ξ ′ ( G ) and the bundle map u ξ : U ξ ( G ) → Ω ξ ( G ) preserveharmonicity. Proposition 3. [1, 4] Let Φ : S \ D → U ξ ( G ) be an extended solution. Thena) u ξ ◦ Φ : S \ D → Ω ξ ( G ) is an extended solution, with ξ ∈ I ( G );b) for each ξ ′ ∈ I ′ ( G ) such that ξ (cid:22) ξ ′ , U ξ,ξ ′ (Φ) = U ξ,ξ ′ ◦ Φ : S \ D → U ξ ′ ( G ) is an extended solution.3.2. Weierstrass representation.
Taking a larger discrete subset if necessary, one obtains a more explicitdescription for harmonic maps of finite uniton number and their extended solutions as follows.
Proposition 4. [1] Let Φ : S → Ω alg G be an extended solution. There exists a discrete set D ′ ⊇ D of S suchthat Φ (cid:12)(cid:12) S \ D ′ = exp C · γ ξ for some holomorphic vector-valued function C : S \ D ′ → u ξ , where u ξ is the finitedimensional nilpotent subalgebra of Λ +alg g C defined by u ξ = M ≤ i 12! (ad C ) C z + . . . + ( − r ( ξ ) − r ( ξ )! (ad C ) r ( ξ ) − C z , (2)the coefficient λ i have zero component in each g ξi +2 , . . . , g ξr ( ξ ) .3.3. S -invariant extended solutions. Extended solutions with values in some Ω ξ ( G ), off a discrete subset,are said to be S -invariant . If we take a unitary representation ρ : G → U ( n ) for some n , then for any suchextended solution Φ we have ρ ◦ Φ λ = P si = r λ i π W i , where, for each i , π W i is the orthogonal projection onto acomplex vector subbundle W i of C n := M × C n and C n = L si = r W i is an orthogonal direct sum decomposition.Set A i = L j ≤ i W j so that { } ⊂ A r ⊂ . . . ⊂ A i − ⊂ A i ⊂ A i +1 ⊂ . . . ⊂ A s = C n . The harmonicity condition amounts to the following conditions on this flag: for each i , A i is a holomorphicsubbundle of C n ; the flag is superhorizontal , in the sense that, for each i , we have ∂A i ⊆ A i +1 , that is, givenany section s of A i then ∂s∂z is a section of A i +1 for any local complex coordinate z of S . Normalization of harmonic maps. Let ∆ := { α , . . . , α r } ⊂ ∆ + be the basis of positive simple roots,with dual basis { H , . . . , H r } ⊂ t , that is α i ( H j ) = i δ ij , where r = rank( g ). Given ξ = P n i H i and ξ ′ = P n ′ i H i in I ′ ( G ), we have n i , n ′ i ≥ ξ (cid:22) ξ ′ if and only if n ′ i ≤ n i for all i . For each I ⊆ { , . . . , r } ,define the cone C I = n P ri =1 n i H i | n i ≥ , n j = 0 iff j / ∈ I o . Definition 1. [6] Let ξ ∈ I ′ ( G ) ∩ C I . We say that ξ is a I -canonical element of G with respect to W if it is amaximal element of ( I ′ ( G ) ∩ C I , (cid:22) ), that is: if ξ (cid:22) ξ ′ and ξ ′ ∈ I ′ ( G ) ∩ C I then ξ = ξ ′ .When G has trivial centre, which is the case considered in [1], there exists a unique I -canonical element,which is given by ξ I = P i ∈ I H i , for each I . When G has non-trivial centre, the I -canonical elements of G weredescribed in [5, 6].Any harmonic map ϕ : S → G admits a normalized extended solution , that is, an extended solution Φ takingvalues in U ξ ( G ), off some discrete set, for some canonical element ξ . This is a consequence of the followinggeneralization of Theorem 4.5 in [1]. Theorem 5. [4] Let Φ : S \ D → U ξ ( G ) be an extended solution. Take ξ ′ ∈ I ′ ( G ) such that ξ (cid:22) ξ ′ and g ξ = g ξ ′ . Then γ − := U ξ,ξ − ξ ′ (Φ) is a constant loop in Ω alg G and γ Φ : S \ D → U ξ ′ ( G ).The uniton number of a normalized extended solution Φ : S \ D → U ξ ( G ) can be computed with respect toany irreducible n -dimensional representation ρ : G → End( V ) with highest weight ω ∗ and lowest weight ̟ ∗ asfollows [6]: r ρ ( ξ ) := ω ∗ ( ξ ) − ̟ ∗ ( ξ ).4. Harmonic spheres in outer symmetric spaces In the following sections we will establish our classification of harmonic maps from S into compact outersymmetric spaces and establish a Weierstrass formula for such harmonic maps. These will allow us to producesome explicit examples of harmonic maps from two-spheres into outer symmetric spaces from meromorphicdata.As we have referred in Section 1, although any harmonic map from a two-sphere into an outer symmetricspace G/K admits a T σ -invariant extended solution, not all T σ -invariant extended solutions correspond toharmonic maps into G/K ; by Proposition 6 and Theorem 10 below, they correspond to a harmonic map intosome possibly different outer symmetric space G/K ′ (compare Theorem 20 with Theorem 23 for an examplewhere this happens).4.1. Symmetric G -spaces and Cartan embeddings. Let N = G/K be a symmetric space, where K is theisotropy subgroup at the base point x ∈ N , and let σ : G → G be the corresponding involution: we have G σ ⊆ K ⊆ G σ , where G σ is the subgroup fixed by σ and G σ denotes its connected component of the identity.We assume that N is a bottom space , i.e. K = G σ . Let g = k σ ⊕ m σ be the ± σ , where k σ is the Lie algebra of K . Consider the (totally geodesic) Cartanembedding ι σ : N ֒ → G defined by ι σ ( g · x ) = gσ ( g − ). The image of the Cartan embedding is precisely theconnected component P σe of P σ := { g ∈ G | σ ( g ) = g − } containing the identity e of the group G . Observe that,given ξ ∈ I ( G ) ∩ k σ , then exp( πξ ) ∈ P σ . We denote by P σξ the connected component of P σ containing exp( πξ ). Proposition 6. Given ξ ∈ I ( G ) ∩ k σ , we have the following.a) G acts transitively on P σξ as follows: for g ∈ G and h ∈ P σξ , g · σ h = ghσ ( g − ) . (3)b) P σξ is a bottom symmetric G -space totally geodesically embedded in G with involution τ = Ad(exp πξ ) ◦ σ. (4)c) For any other ξ ′ ∈ I ( G ) ∩ k σ we have exp( πξ ′ ) ∈ P τ and P τξ ′ = exp( πξ ) P σξ ′ − ξ . N. CORREIA AND R. PACHECO d) The ± g = k τ ⊕ m τ associated to the symmetric G -space P σξ at the fixedpoint exp( πξ ) ∈ P σξ is given by k C τ = M g ξ i ∩ k C σ ⊕ M g ξ i +1 ∩ m C σ (5) m C τ = M g ξ i +1 ∩ k C σ ⊕ M g ξ i ∩ m C σ . (6) Proof. Take h ∈ P σ . We have σ ( g · σ h ) = σ ( ghσ ( g − )) = σ ( g ) h − g − = ( ghσ ( g − )) − = ( g · σ h ) − . Then g · σ h ∈ P σ and we have a continuous action of G on P σ . Since G is connected, this action induces anaction of G on each connected component of P σ . Since g · σ e = gσ ( g − ) = ι σ ( g · x ) and ι σ ( N ) = P σe , the action · σ of G on P σe is transitive.Take ξ ∈ I ( G ) ∩ k σ , so that σ ( ξ ) = ξ and exp 2 πξ = e . Consider the involution τ defined by (4). If g ∈ P σ ,then τ (exp( πξ ) g ) = exp( πξ ) σ (exp( πξ ) g ) exp( πξ ) = σ ( g ) exp( πξ ) = (exp( πξ ) g ) − , which means that exp( πξ ) g ∈ P τ . Reciprocally, if exp( πξ ) g ∈ P τ , one can check similarly that g ∈ P σ . Hence P τ = exp( πξ ) P σ . In particular, by continuity, P τξ ′ = exp( πξ ) P σξ ′ − ξ for any other ξ ′ ∈ I ( G ) with σ ( ξ ′ ) = ξ ′ .Reversing the rules of σ = Ad(exp πξ ) ◦ τ and τ , we also have P σξ = exp( πξ ) P τe . Since G acts transitively on P τe , for each h ∈ P σξ there exists g ∈ G such that h = exp( πξ )( g · τ e ) = (exp( πξ ) g ) · σ exp( πξ ) . This shows that G also acts transitively on P σξ . The isotropy subgroup at exp( πξ ) consists of those elements g of G satisfying g exp( πξ ) σ ( g − ) = exp( πξ ), that is those elements g of G which are fixed by τ :exp( πξ ) σ ( g ) exp( πξ ) = g. (7)Hence P σξ ∼ = G/G τ , which is a bottom symmetric G -space with involution τ . Since P τe ⊂ G totally geodesicallyand P σξ is the image of P τe under an isometry (left multiplication by exp πξ ), then P σξ ⊂ G totally geodesically.Differentiating (7) at the identity we get k τ = { X ∈ g | X = Ad(exp πξ ) ◦ σ ( X ) } . Taking account of theformula Ad(exp( πξ )) = e π ad ξ and that σ commutes with ad ξ , we obtain (5); and (6) follows similarly. (cid:3) Outer symmetric spaces. The existence of outer involutions of a simple Lie algebra g depends on theexistence of non-trivial involutions of the Dynkin diagram of g C [2, 8, 12, 14]. Fix a maximal abelian subalgebra t of g and a Weyl chamber W in t , which amounts to fix a system of positive simple roots ∆ = { α , . . . , α r } ,where r = rank( g ). Let ̺ be a non-trivial involution of the Dynkin diagram and σ ̺ the fundamental outerinvolution associated to ̺ [2, 14]. The (local isometry classes of) outer symmetric spaces of compact typeassociated to involutions of the form σ ̺ are precisely SU (2 n ) /Sp ( n ), SU (2 n + 1) /SO (2 n + 1), E /F and the real projective spaces R P n − .These spaces are called the fundamental outer symmetric spaces . The remaining classes of outer involutions areobtained as follows [1, 14].Let g = k ̺ ⊕ m ̺ be the corresponding ± g . As shown in Proposition 3.20 of[2], the Lie subalgebra k ̺ is simple and the orthogonal projection of ∆ onto k ̺ , π k ̺ (∆ ), is a basis of positivesimple roots of k ̺ associated to the maximal abelian subalgebra t k ̺ := t ∩ k ̺ . Consider the split t = t k ̺ ⊕ t m ̺ with respect to g = k ̺ ⊕ m ̺ . Set s = r − k , where k = rank( k ̺ ). We can label the basis ∆ in order to getthe following relations: ̺ ( α j ) = α j for 1 ≤ j ≤ k − s and ̺ ( α j ) = α s + j for k − s + 1 ≤ j ≤ k . Let π k ̺ be theorthogonal projection of t onto t k ̺ , that is π k ̺ ( H ) = ( H + σ ̺ ( H )) for all H ∈ t . Set π k ̺ (∆ ) = { β , . . . , β k } ,with β j = (cid:26) α j for 1 ≤ j ≤ k − s ( α j + α j + s ) for k − s + 1 ≤ j ≤ k . (8) This is a basis of i t ∗ k ̺ with dual basis { ζ , . . . , ζ k } given by ζ j = (cid:26) H j for 1 ≤ j ≤ k − sH j + H j + s for k − s + 1 ≤ j ≤ k . (9) Theorem 7. [14] Let ̺ be an involution of the Dynkin diagram of g . Let ω = k − s X j =1 n j β j + k X j = k − s +1 n ′ j β j be the highest root of k ̺ with respect to π k ̺ (∆ ) = { β , . . . , β k } , defined as in (8). Given i such that n i = 1 or2, define an involution σ ̺,i by σ ̺,i = Ad(exp πζ i ) ◦ σ ̺ . (10)Then any outer involution of g is conjugate in Aut ( g ), the group of automorphisms of g , to some σ ̺ or σ ̺,i . Inparticular, there are at most k − s + 1 conjugacy classes of outer involutions.The list of all (local isometry classes of) irreducible outer symmetric spaces of compact type is shown inTable 1 (cf. [2, 8, 12]). G/K rank( G ) rank( K ) rank( G/K ) dim( G/K ) SU (2 n ) /SO (2 n ) 2 n − n n − n − n + 1) SU (2 n + 1) /SO (2 n + 1) 2 n n n n (2 n + 3) SU (2 n ) /Sp ( n ) 2 n − n n − n − n + 1) G p ( R n ) ( p odd ≤ n ) n n − p p (2 n − p ) E /Sp (4) 6 4 6 42 E /F Table 1. Irreducible outer symmetric spaces.Given an outer involution σ of the form σ ̺,i or σ ̺ and its ± g = k σ ⊕ m σ , set t k σ = t ∩ k σ , which is a maximal abelian subalgebra of k σ . Following [8], a non-empty intersection of t k σ witha Weyl chamber in t is called a compartment . Each compartment lies in a Weyl chamber in t k σ and the Weylchambers in t k σ can be decomposed into the same number of compartments [8].The intersection of the integer lattice I ( G ) with the Weyl chamber W in t , which we have denoted by I ′ ( G ),is described in terms of the dual basis { H , . . . , H r } ⊂ t , with r = rank( g ), by I ′ ( G ) = (cid:8) r X i =1 n i H i ∈ I ( G ) | n i ∈ N for all i (cid:9) . When σ is a fundamental outer involution σ ̺ , the compartment W ∩ t k ̺ is itself a Weyl chamber in t k ̺ . Then,the intersection of the integer lattice I ( G σ ̺ ) with the Weyl chamber W ∩ t k ̺ , is given by I ′ ( G σ ̺ ) = (cid:8) k X i =1 n i ζ i ∈ I ( G ) | n i ∈ N for all i (cid:9) = I ′ ( G ) ∩ t k ̺ . Cartan embeddings of fundamental outer symmetric spaces. Next we describe those elements ξ of I ′ ( G σ ̺ )for which the connected component P σ ̺ ξ of P σ ̺ containing exp( πξ ) can be identified with the fundamental outersymmetric G -space associated to ̺ . Start by considering the following σ ̺ -invariant subsets of the root system∆ ⊂ i t ∗ of g : ∆( k ̺ ) = { α ∈ ∆ | g α ⊂ k C ̺ } , ∆( m ̺ ) = { α ∈ ∆ | g α ⊂ m C ̺ } , ∆ ̺ = ∆ \ (∆( k ̺ ) ∪ ∆( m ̺ )) . (11) N. CORREIA AND R. PACHECO Then k C ̺ = t C k ̺ ⊕ π k ̺ ( r ̺ ) ⊕ M α ∈ ∆( k ̺ ) g α , m C ̺ = t C m ̺ ⊕ π m ̺ ( r ̺ ) ⊕ M α ∈ ∆( m ̺ ) g α , where r ̺ = L α ∈ ∆ ̺ g α . Since the involution ̺ acts on ∆ ̺ as a permutation without fixed points, we can fixsome subset ∆ ′ ̺ so that ∆ ̺ is the disjoint union of ∆ ′ ̺ with ̺ (∆ ′ ̺ ):∆ ̺ = ∆ ′ ̺ ⊔ ̺ (∆ ′ ̺ ) . (12)For each α ∈ ∆ ′ ̺ , σ ̺ restricts to an involution in the subspace g α ⊕ g ̺ ( α ) ⊂ r ̺ . Hence we have the following. Lemma 8. The orthogonal projections of r ̺ onto k C ̺ and m C ̺ are given by π k ̺ ( r ̺ ) = M α ∈ ∆ ′ ̺ k C ̺ ∩ (cid:0) g α ⊕ g ̺ ( α ) (cid:1) , π m ̺ ( r ̺ ) = M α ∈ ∆ ′ ̺ m C ̺ ∩ (cid:0) g α ⊕ g ̺ ( α ) (cid:1) , and, for each α ∈ ∆ ′ ̺ , k C ̺ ∩ (cid:0) g α ⊕ g σ ̺ ( α ) (cid:1) = { X α + σ ̺ ( X α ) | X α ∈ g α } , m C ̺ ∩ (cid:0) g α ⊕ g σ ( α ) (cid:1) = { X α − σ ̺ ( X α ) | X α ∈ g α } . In particular, dim r ̺ = 2 dim π k ̺ ( r ̺ ) = 2 dim π m ̺ ( r ̺ ). Proposition 9. Consider the dual basis { ζ , . . . , ζ k } defined by (9). Given ξ ∈ I ′ ( G σ ̺ ) with ξ = P ki =1 n i ζ i and n i ≥ 0, then P σ ̺ ξ is a fundamental outer symmetric space with involution (conjugated to) σ ̺ if and only if n i is even for each 1 ≤ i ≤ k − s . Proof. There is only one class of outer symmetric SU (2 n + 1)-spaces and, in this case, the involution ̺ doesnot fix any simple root, that is k − s = 0. Hence the result trivially holds for N = SU (2 n + 1) /SO (2 n + 1) . Next we consider the remaining fundamental outer symmetric spaces, which are precisely the symmetricspaces of rank-split type [8], those satisfying ∆( m ̺ ) = ∅ . For such symmetric spaces, the reductive symmetricterm m ̺ satisfies m ̺ = t m ̺ ⊕ π m ̺ ( r ̺ ). On the other hand, in view of (6), we have, for τ = Ad(exp πξ ) ◦ σ ̺ , m C τ = M g ξ i +1 ∩ k C ̺ ⊕ M g ξ i ∩ m C ̺ = t C m ̺ ⊕ M α ∈ ∆( k ̺ ) ∩ ∆ − ξ g α ⊕ M α ∈ ∆ ′ ̺ ∩ ∆ − ξ k C ̺ ∩ ( g α ⊕ g ̺ ( α ) ) ⊕ M α ∈ ∆ ′ ̺ ∩ ∆ + ξ m C ̺ ∩ ( g α ⊕ g ̺ ( α ) ) , where ∆ + ξ := { α ∈ ∆ | α ( ξ )i is even } and ∆ − ξ := { α ∈ ∆ | α ( ξ )i is odd } . Taking into account Lemma 8, fromthis we see that dim m τ = dim m ̺ (which means, by Table 1, that P σ ̺ ξ is a fundamental outer symmetric spacewith involution conjugated to σ ̺ ) if and only if M α ∈ ∆( k ̺ ) ∩ ∆ − ξ g α = { } , which holds if and only if ξ = P ki =1 n i ζ i with n i even for each 1 ≤ i ≤ k − s . (cid:3) Harmonic spheres in symmetric G -spaces. Given an involution σ on G , define an involution T σ onΩ G by T σ ( γ )( λ ) = σ ( γ ( − λ ) γ ( − − ) . Let Ω σ G be the fixed set of T σ . Theorem 10. [8, 11] Given ξ ∈ I ( G ) ∩ k σ , any harmonic map ϕ : S → P σξ ⊂ G admits an T σ -invariantextended solution Φ : S → Ω σ G . Conversely, given an T σ -invariant extended solution Φ, the smooth map ϕ = Φ − from S is harmonic and takes values in some connected component of P σ . Proposition 11. [8] Given Φ ∈ U σξ ( G ) := U ξ ( G ) ∩ Ω σ G , with ξ ∈ I ( G ) ∩ k σ , set γ = u ξ ◦ Φ. Then γ takesvalues in K . By continuity, Φ − and γ ( − 1) take values in the same connected component of P σ .Hence, together with Theorems 1 and 10, this implies the following. Theorem 12. Any harmonic map ϕ from S into a connected component of P σ admits an extended solutionΦ : S \ D → U σξ ( G ) := U ξ ( G ) ∩ Ω σ G , for some ξ ∈ I ′ ( G ) ∩ k σ and some discrete subset D . If σ = σ ̺ is thefundamental outer involution, then ϕ = Φ − takes values in P σ ̺ ξ . Proof. By Proposition 11, Φ and γ := u ξ ◦ Φ take values in the same connected component of P σ when evaluatedat λ = − 1. Since γ : S → G σ is a homomorphism, γ is in the G σ -conjugacy class of γ ξ ′ for some ξ ′ ∈ I ′ ( G σ ),where G σ is the subgroup of G fixed by σ . Consequently, γ ( − 1) = gγ ξ ′ ( − g − = g · σ γ ξ ′ ( − , for some g ∈ G σ ,which means that γ ( − 1) takes values in the connected component P σξ ′ . On the other hand, γ is in the G -conjugacy class of γ ξ , with ξ ∈ I ′ ( G ) ∩ k σ . If σ is the fundamental outer involution σ ̺ , then I ′ ( G σ ) = I ′ ( G ) ∩ k σ ;and we must have ξ = ξ ′ . (cid:3) Remark 1. If σ is not a fundamental outer involution, each Weyl chamber W σ in t k σ can be decomposed intomore than one compartment: W σ = C ⊔ . . . ⊔ C l , where C = W ∩ t k σ and the remaining compartments areconjugate to C under G [8], that is, there exists g i ∈ G satisfying C i = Ad( g i )( C ) for each i . Hence, if wehave an extended solution Φ : S \ D → U σξ ( G ) with ξ ∈ I ′ ( G ) ∩ k σ ⊂ C , the corresponding harmonic mapΦ − takes values in one of the connected components P σg i ξg − i .4.2.1. ̺ -canonical elements. Let I be a subset of { , . . . , k } , with k = rank( k ̺ ), and set C ̺I = n k X i =1 n i ζ i | n i ≥ , n j = 0 iff j / ∈ I o . Let ξ ∈ I ′ ( G σ ̺ ) ∩ C ̺I . We say that ζ is a ̺ -canonical element of G (with respect to the choice of W ) if ζ is amaximal element of ( I ′ ( G σ ̺ ) ∩ C ̺I , (cid:22) ), that is: if ζ (cid:22) ζ ′ and ζ ′ ∈ I ′ ( G σ ̺ ) ∩ C ̺I then ζ = ζ ′ . Remark 2. When G has trivial centre, the duals ζ , . . . , ζ k belong to the integer lattice. Then, for each I thereexists a unique ̺ -canonical element, which is given by ζ I = P i ∈ I ζ i . In this case, our definition of ̺ -canonicalelement coincides with that of S -canonical element in [8].Now, consider a fundamental outer involution σ ̺ and let N be an associated outer symmetric G -space, thatis, N corresponds to an involution of G of the form σ ̺ or σ ̺,i , with ζ i in the conditions of Theorem 7. If G has trivial centre, we certainly have ζ i ∈ I ′ ( G σ ̺ ). As a matter of fact, as we will see later, in most cases wehave ζ i ∈ I ′ ( G σ ̺ ), whether G has trivial centre or not, with essentially one exception: for G = SU (2 n ) and N = SU (2 n ) /SO (2 n ). So, we will treat this case separately and assume henceforth that ζ i ∈ I ′ ( G σ ̺ ). Remark 3. Consider the Dynkin diagram of e : α α α α α α b b b b bb This admits a unique nontrivial involution ̺ . Let { H , . . . , H } be the dual basis of ∆ = { α , . . . , α } . Thesemi-fundamental basis π k ̺ (∆ ) = { β , β , β , β } is given by β = α , β = α , β = α + α and β = α + α ,whereas the dual basis is given by ζ = H , ζ = H , ζ = H + H and ζ = H + H . Taking account thatthe elements H i are related with the duals η i of the fundamental weights by[ H i ] = / / / / 31 2 2 3 2 15 / / / / 32 3 4 6 4 24 / / / / / / / / [ η i ] , we see that the elements ζ i are in the integer lattice I ′ ( ˜ E ) ⊂ I ′ ( E ), where ˜ E is the compact simply connectedLie group with Lie algebra e , which has centre Z , and E is the adjoint group ˜ E / Z .Taking into account Proposition 6, we can identify N with the connected component P σ ̺ ζ i = exp( πζ i ) P σ ̺,i e ,which is a totally geodesic submanifold of G , via g · x ∈ N exp( πζ i ) gσ ̺,i ( g − ) ∈ P σ ̺ ζ i . (13)By Theorem 12, each harmonic map ϕ : S → N ∼ = P σ ̺ ζ i admits a T σ ̺ -invariant extended solution with values,off a discrete set, in some unstable manifold U ξ ( G ), with ξ ∈ I ′ ( G σ ̺ ) ∩ C ̺I . By Theorem 5, this extended solutioncan be multiplied on the left by a constant loop in order to get a normalized extended solution with valuesin some unstable manifold U ζ ( G ) for some ̺ -canonical element ζ . Hence, if G has trivial centre, the Bruhatdecomposition of Ω alg G gives rise to 2 k classes of harmonic maps into P σ ̺ , that is 2 k classes of harmonic mapsinto all outer symmetric G -spaces.However, the normalization procedure given by Theorem 5 does not preserve T σ ̺ -invariance, and conse-quently, as we will see next, normalized extended solutions with values in the same unstable manifold U ζ ( G ),for some ̺ -canonical element ζ , correspond in general to harmonic maps into different outer symmetric G -spaces.Hence the classification of harmonic two-spheres into outer symmetric G -spaces in terms of ̺ -canonical elementsis manifestly unsatisfactory since it does not distinguishes the underlying symmetric space. In the followingsections we overcome this weakness by establishing a classification of all such harmonic maps in terms of pairs( ζ, σ ), where ζ is a ̺ -canonical element and σ an outer involution of G .4.2.2. Normalization of T σ -invariant extended solutions. Let σ be an outer involution of G . The fibre bundlemorphisms U ξ,ξ ′ preserve T σ -invariance: Proposition 13. If ξ (cid:22) ξ ′ and ξ, ξ ′ ∈ I ′ ( G ) ∩ k σ , then U ξ,ξ ′ ( U σξ ( G )) ⊂ U σξ ′ ( G ). Proof. For Φ ∈ U σξ ( G ), write Φ = Ψ · γ ξ for some Ψ ∈ Λ +alg G C . If Φ is T σ -invariant we have Ψ( λ ) · γ ξ = σ (Ψ( − λ )) · γ ξ . Consequently, we also have Ψ( λ ) · γ ξ ′ = σ (Ψ( − λ )) · γ ξ ′ , which means in turn that U ξ,ξ ′ (Φ) = Ψ · γ ′ ξ is T σ -invariant. (cid:3) Hence, if Φ : S \ D → U σξ ( G ) is an extended solution and ξ (cid:22) ξ ′ , with ξ, ξ ′ ∈ I ′ ( G ) ∩ k ̺ , by Theorem 5 andProposition 13 we know that γ − := U ξ,ξ − ξ ′ (Φ) is a constant T σ -invariant loop if g ξ = g ξ ′ . However, in general,the product γ Φ is not T σ -invariant. Lemma 14. Assume that γ − , Φ ∈ Ω σ G and γ ( − ∈ P σξ for some ξ ∈ I ( G ) ∩ k σ . Take h ∈ G such that γ ( − 1) = h − · σ exp( πξ ). Then hγ Φ h − ∈ Ω τ G , with τ = Ad(exp πξ ) ◦ σ . Proof. Since γ − , Φ ∈ Ω σ G , a simple computation shows that T σ ( γ Φ) = γ ( − − γ Φ γ ( − . Since γ ( − ∈ P σξ ,there exists h ∈ G such that γ ( − 1) = h − · σ exp( πξ ) = h − exp( πξ ) σ ( h ). One can check now that T τ ( hγ Φ h − ) = hγ Φ h − . (cid:3) Proposition 15. Take ξ, ξ ′ ∈ I ′ ( G ) ∩ k σ such that ξ (cid:22) ξ ′ . Let Φ : S \ D → U σξ ( G ) be a T σ -invariant extendedsolution. If γ − := U ξ,ξ − ξ ′ (Φ) is a constant loop, there exists h ∈ G such that ˜Φ := hγ Φ h − takes values in U τξ ′ ( G ), with τ = Ad(exp π ( ξ − ξ ′ )) ◦ σ. Additionally, if σ is the fundamental outer involution σ ̺ , the harmonic map Φ − takes values in P σξ and ˜Φ − takes values in P τξ ′ , which implies that Φ − is given, up to isometry, byexp( π ( ξ − ξ ′ )) ˜Φ − : S → P σξ . Proof. Assume that γ − := U ξ,ξ − ξ ′ (Φ) = Ψ · γ ξ − ξ ′ is a constant loop. We can write Ψ γ ξ − ξ ′ = γ − b for some b : S \ D → Λ +alg G . Then Φ = Ψ · γ ξ = Ψ · γ ξ − ξ ′ γ ξ ′ = γ − b · γ ξ ′ , which implies that γ Φ takes values in U ξ ′ ( G ). On the other hand, since γ − is T σ -invariant (by Proposition 13), γ ( − ∈ P σ .Take η ∈ I ′ ( G σ ) and h ∈ G such that γ ( − ∈ P ση and γ ( − 1) = h − · σ exp πη . From Lemma 14, we see that˜Φ := hγ Φ h − is T τ -invariant. Hence ˜Φ takes values in U τξ ′ ( G ). Since γ is constant, ˜Φ is an extended solution.If σ = σ ̺ , then I ′ ( G σ ̺ ) = I ′ ( G ) ∩ k σ ̺ , which implies that η = ξ − ξ ′ . The element h ∈ G is such that γ ( − 1) = h − exp( π ( ξ − ξ ′ )) σ ̺ ( h ) . On the other hand, since, by Theorem 12, Φ − takes values in P σ ̺ ξ , we also have Φ − = g exp( πξ ) σ ̺ ( g − ) forsome lift g : S → G . Hence˜Φ − = hγ ( − − h − = exp( π ( ξ − ξ ′ )) σ ̺ ( h ) g exp( πξ ) σ ̺ ( σ ̺ ( h ) g ) − = exp( π ( ξ − ξ ′ ))( σ ̺ ( h ) g · σ ̺ exp πξ )Hence, in view of Proposition 6, ˜Φ − takes values in P τξ ′ = exp( π ( ξ − ξ ′ )) P σξ . (cid:3) Under some conditions on ξ (cid:22) ξ ′ , the morphism U ξ,ξ − ξ ′ (Φ) is always a constant loop. Proposition 16. Take ξ, ξ ′ ∈ I ′ ( G ) ∩ k σ such that ξ (cid:22) ξ ′ . Assume that g ξ i ∩ m C σ ⊂ M ≤ j< i g ξ − ξ ′ j , g ξ i − ∩ k C σ ⊂ M ≤ j< i − g ξ − ξ ′ j , (14)for all i > 0. Then, U ξ,ξ − ξ ′ : U σξ ( G ) → U σξ − ξ ′ ( G ) transforms T σ -invariant extended solutions in constant loops. Proof. Given an extended solution Φ : S \ D → U σξ ( G ), choose Ψ : S \ D → Λ +alg G C such that Φ = Ψ · γ ξ and T σ (Ψ) = Ψ. Differentiating this we see thatImΨ − Ψ z ⊂ M i ≥ λ i k C σ ⊕ M i ≥ λ i +1 m C σ . (15)Write Ψ − Ψ z = P r ≥ λ r X ′ r . Since ξ (cid:22) ξ − ξ ′ , by Proposition 3 and Proposition 13, U ξ,ξ − ξ ′ (Φ) is an extendedsolution with values in U σξ − ξ ′ ( G ). Hence, taking into account Lemma 2, in order to prove that U ξ,ξ − ξ ′ (Φ) isconstant we only have to check that the component of X ′ r over g ξ − ξ ′ r +1 vanishes for all r ≥ r = 2 i , X ′ i takes values in L j ≤ i +1 g ξj ∩ k C σ . But, since ξ (cid:22) ξ − ξ ′ and, byhypothesis, (14) holds, we have M j ≤ i +1 g ξj ∩ k C σ = (cid:0) M j ≤ i g ξj ∩ k C σ (cid:1) ⊕ (cid:0) g ξ i +1 ∩ k C σ (cid:1) ⊂ (cid:0) M j ≤ i g ξ − ξ ′ j ∩ k C σ (cid:1) ⊕ M ≤ j< i +1 g ξ − ξ ′ j . Hence the component of X ′ i over g ξ − ξ ′ i +1 vanishes for all i ≥ 0. Similarly, for r = 2 i − X ′ i − takes values in L j ≤ i g ξj ∩ m C σ , and we can check that the component of X ′ i − over g ξ − ξ ′ i vanishes for all i > γ − := U ξ,ξ − ξ ′ (Φ) = Ψ · γ ξ − ξ ′ is a constant loop. (cid:3) Definition 2. We say that ζ ∈ I ′ ( G σ ̺ ) ∩ C ̺I is a ̺ - semi-canonical element if ζ is of the form ζ = P i ∈ I n i ζ i with 1 ≤ n i ≤ m i , where m i is the least positive integer which makes m i ζ i ∈ I ′ ( G σ ̺ ). Corollary 1. Take ξ ∈ I ′ ( G σ ̺ ) ∩ C ̺I , with I ⊂ { , . . . , k } . Let Φ : S \ D → U σ ̺ ξ ( G ) be a T σ ̺ -invariant extendedsolution, and let ϕ : S → P σ ̺ ξ be the corresponding harmonic map. Then there exist h ∈ G , a constant loop γ , and a ̺ - semi-canonical ζ such that ˜Φ := hγ Φ h − defined on S \ D takes values in U σ ̺ ζ ( G ). The harmonicmap ˜Φ − takes values in P σ ̺ ζ = P σ ̺ ξ and coincides with ϕ up to isometry. Proof. Write ξ = P i ∈ I r i ζ i , with r i > 0. For each i ∈ I , let n i be the unique integer number in { , . . . , m i } such that n i = r i mod 2 m i . Set ζ = P i ∈ I n i ζ i . It is clear that ξ (cid:22) ζ and ζ ∈ I ′ ( G σ ̺ ) ∩ C ̺I . Observe also thatconditions (14) hold automatically for any ξ ′ ∈ I ′ ( G σ ̺ ) ∩ C ̺I satisfying ξ (cid:22) ξ ′ . In particular they hold for ξ ′ = ζ .Finally, since ξ − ζ = 2 P i ∈ I m i k i ζ i for some nonnegative integer numbers k i , then exp π ( ξ − ζ ) = e , and theresult follows from Propositions 15 and 16. (cid:3) Classification of harmonic two-spheres into outer symmetric spaces. To sum up, in order to classify allharmonic two-spheres into outer symmetric spaces we proceed as follows:(1) Start with a fundamental outer involution σ ̺ and let N be an outer symmetric G -space correspondingto an involution of the form σ ̺ or σ ̺,i of G , according to (10), where the element ζ i is in the conditionsof Theorem 7. We assume that exp 2 πζ i = e , that is ζ i ∈ I ′ ( G σ ̺ ). Let ϕ : S → N be an harmonicmap and identify N with P σ ̺ ζ i = exp( πζ i ) P σ ̺,i e via the totally geodesic embedding (13). If N is thefundamental outer space with involution σ ̺ we simply identify N with P σ ̺ e via ι σ ̺ .(2) By Theorem 12, ϕ : S → N ∼ = P σ ̺ ζ i admits a T σ ̺ -invariant extended solution Φ : S → Ω σ ̺ G whichtakes values, off some discrete subset D , in some unstable manifold U σ ̺ ζ ′ ( G ), with ζ ′ ∈ I ′ ( G σ ̺ ); moreover, P σ ̺ ζ ′ = P σ ̺ ζ i .(3) By Corollary 1, we can assume that ζ ′ is a ̺ -semi-canonical element in I ′ ( G σ ̺ ) ∩ C ̺I . If ζ is a ̺ -canonicalelement such that ζ ′ (cid:22) ζ and U ζ ′ ,ζ ′ − ζ (Φ) is constant, then, taking into account Proposition 15, thereexists a T τ -invariant extended solution ˜Φ : S \ D → U τζ ( G ) , where τ = Ad(exp π ( ζ ′ − ζ )) ◦ σ ̺ , (16)such that the harmonic map ϕ is given, up to isometry, by ˜Φ − : S → P τζ . Here we identify N with P τζ = exp( π ( ζ ′ − ζ )) P σ ̺ ζ i via the composition of (13) with the left multiplication by exp( π ( ζ ′ − ζ )).(4) By Proposition 16, there always exists a ̺ -canonical element ζ in such conditions.Hence, we classify harmonic spheres into outer symmetric G -spaces in terms of pairs ( ζ, τ ), where ζ is a ̺ -canonical element and τ is an outer involution of the form (16) for some ̺ -semi-canonical element ζ ′ with ζ ′ (cid:22) ζ .4.2.4. Weierstrass Representation for T σ -invariant Extended Solutions. From (15) and Proposition 4, we obtainthe following. Theorem 17. Let Φ : M → Ω σ alg G be an extended solution. There exists a discrete set D ′ ⊇ D of M such thatΦ (cid:12)(cid:12) M \ D ′ = exp C · γ ξ for some holomorphic vector-valued function C : M \ D ′ → ( u ξ ) σ , where ( u ξ ) σ is the finitedimensional nilpotent subalgebra of Λ +alg g C defined by( u ξ ) σ = M ≤ i Next we will describe explicit examples of harmonic spheres into classical outer symmetric spaces.5.1. Outer symmetric SO (2 n ) -spaces. For details on the structure of so (2 n ) see [10]. Consider on R n thestandard inner product h· , ·i and fix a complex basis u = { u , . . . , u n , u , . . . , u n } of C n = ( R n ) C satisfying h u i , u j i = 0 , h u i , u j i = δ ij , for all 1 ≤ i, j ≤ n. Throughout this section we will denote by V l the l -dimensional isotropic subspace spanned by u , . . . , u l .Set E i = E i,i − E n + i,n + i , where E j,j is a square matrix, with respect to the basis u , whose ( j, j )-entry is iand all other entries are 0. The complexification t C of the algebra t of diagonal matrices P a i E i , with a i ∈ R and P a i = 0, is a Cartan subalgebra of so (2 n ) C . Let { L , . . . , L n } be the dual basis in i t ∗ of { E , . . . , E n } ,that is L i ( E j ) = i δ ij . The roots of so (2 n ) are the vectors ± L i ± L j and ± L i ∓ L j , with i = j and 1 ≤ i, j ≤ n .Consider the endomorphisms X i,j = E i,j − E n + j,n + i , Y i,j = E i,n + j − E j,n + i , Z i,j = E n + i,j − E n + j,i , (17)where E i,j , with i = j , is a square matrix whose ( i, j )-entry is 1 and all other entries are 0. The root spaces of L i − L j , L i + L j and − L i − L j , respectively, are generated by the endomorphisms X i,j , Y i,j and Z i,j , respectively.Fix the positive root system ∆ + = { L i ± L j } i 2, and β n − = ( α n − + α n ) = L n − , whereas the dual basis { ζ , . . . , ζ n − } is given by ζ i = E + . . . + E i , with i = 1 , . . . , n − 1. Since each ζ i belongsto the integer lattice I ( SO (2 n ) σ ̺ ), we have: Proposition 18. The ̺ -semi-canonical elements of SO (2 n ) are precisely the elements ζ = P n − i =1 m i ζ i suchthat m i ∈ { , , } for 1 ≤ i ≤ n − SO (2 n )-space is the real projective space R P n − , and the associatedouter symmetric SO (2 n )-spaces are the real Grassmannians G p ( R n ) with p > Harmonic maps into real projective spaces R P n − . Consider as base point the one dimensional realvector space V spanned by e n = ( u n + u n ) / √ R n , which establishes an identification of R P n − with SO (2 n ) /S ( O (1) O (2 n − . Denote by π V and π ⊥ V the orthogonal projections onto V and V ⊥ , respectively.The fundamental involution is given by σ ̺ = Ad( s ), where s = π V − π ⊥ V . Following the classificationprocedure established in Section 4.2.3, we start by identifying R P n − with P σ ̺ e . Theorem 19. Each harmonic map ϕ : S → R P n − belongs to one of the following classes: ( ζ l , σ ̺,l ), with1 ≤ l ≤ n − Proof. Let ζ be a ̺ -semi-canonical element and write ζ = X i ∈ I ζ i + X i ∈ I ζ i (18)for some disjoint subsets I and I of { , . . . , n − } . By Proposition 9, P σ ̺ ζ ∼ = R P n − if and only if either I = ∅ or I = { n − } . Suppose that I = { n − } . In this case, exp πζ = exp πζ n − ∈ P σ ̺ ζ n − . We claim that P σ ̺ ζ n − is not the connected component of P σ ̺ containing the identity e . Write exp πζ n − = π V − π ⊥ V , where V isthe two-dimensional real space spanned by e n and e n . For each g ∈ P σ ̺ e , since the G -action · σ ̺ defined by (3)is transitive, we have g = h · σ ̺ e = hs h − s for some h ∈ G , which means that gs = hs h − . In particular, the+1-eigenspaces of gs must be 1-dimensional. However, a simple computation shows that the +1-eigenspace ofexp( πζ n − ) s is 3-dimensional, which establishes our claim. Then, any harmonic map ϕ : S → R P n − ∼ = P σ ̺ e admits a T σ ̺ -invariant extended solution Φ : S \ D → U σ ̺ ζ ( SO (2 n )) with ζ a ̺ -semi-canonical element of the form ζ = P i ∈ I ζ i . Set l = max I . Next we check that ζ and ζ l satisfy the conditions of Proposition 16, with ξ = ζ and ξ ′ = ζ l . It is clear that ζ (cid:22) ζ l . Now, accordingto (11) and (12), we can take ∆ ′ ̺ = { L i − L n , L n − L i } . Hence, for i > g ζ i ∩ m C ̺ = M α ∈ ∆ ′ ̺ ∩ ∆ iζ ( g α ⊕ g ̺ ( α ) ) ∩ m C ̺ , where ∆ iζ = { α ∈ ∆ | α ( ζ ) = 2 i i } . Since( L j − L n )( ζ ) = ( α j + α j +1 + . . . + α n − )( ζ ) = 2 | I ∩ { j, . . . , n − }| i , we have ∆ ′ ̺ ∩ ∆ iζ = { L j − L n | ≤ j ≤ l , and | I ∩ { j, . . . , l }| = i } . Then, given a root α = L j − L n ∈ ∆ ′ ̺ ∩ ∆ iζ (in particular, j ≤ l ) we have α ( ζ − ζ l ) = (2 i − , which meansthat g α ⊂ g ζ − ζ l i − . Consequently, g ζ i ∩ m C ̺ ⊂ M ≤ j< i g ζ − ζ l j . Since g ζ i − = { } for all i , we conclude that (14) holds, and the statement follows from Propositions 15 and16. (cid:3) It is known [3] that there are no full harmonic maps ϕ : S → R P n − . The class of harmonic maps associatedto ( ζ l , σ ̺,l ) consists precisely of those ϕ with ϕ ( S ) contained, up to isometry, in some R P l , as shown in thenext theorem. Theorem 20. Given 1 ≤ l ≤ n − 1, any harmonic map ϕ : S → R P n − in the class ( ζ l , σ ̺,l ) is given by ϕ = R ∩ ( A ⊕ A ) ⊥ , (19)where R is a constant 2 l + 1-dimensional subspace of R n and A is a holomorphic isotropic subbundle of S × R of rank l satisfying ∂A ⊆ A ⊥ . The corresponding extended solutions have uniton number 2 with respect to thestandard representation of SO (2 n ). Proof. Let ϕ : S → R P n − be a harmonic map in the class ( ζ l , σ ̺,l ). This means that ϕ admits an extendedsolution Φ : S \ D → U σ ̺,l ζ l ( SO (2 n )). Up to isometry, ϕ is given by Φ − , which takes values in P σ ̺,l ζ l =exp( πζ l ) P σ ̺ e . This connected component is identified with R P n − via g · V exp( πζ l ) gσ ̺ ( g − ) . (20)Write γ ζ l ( λ ) = λ − π V l + π ⊥ V l ⊕ V l + λπ V l , where V l is the l -dimensional isotropic subspace spanned by u , . . . , u l .We have r ( ζ l ) = 2 if l > r ( ζ ) = 1. Consequently, by Theorem 17,( u ζ l ) σ ̺,l = ( p ζ l ) ⊥ ∩ k C σ ̺,l ⊕ λ ( p ζ l ) ⊥ ∩ m C σ ̺,l . Here ( p ζ l ) ⊥ = g ζ l , which is the null space for l = 1. For l > 1, since ζ l = E + . . . + E l , we have g ζ l = { L i + L j | ≤ i < j ≤ l } ⊂ ∆( k ̺ ) and, from (6), m C σ ̺,l = M g ζ l i +1 ∩ k C ̺ ⊕ M g ζ l i ∩ m C ̺ . Hence ( p ζ l ) ⊥ ∩ m C σ ̺,l = g ζ l ∩ m C ̺ = { } . Then, for any l ≥ 1, we can write Φ = exp C · γ ζ l for some holomorphicfunction C : S \ D → ( p ζ l ) ⊥ ∩ k C σ ̺,l = ( g ζ l ⊕ g ζ l ) ∩ k C σ ̺,l , which means that Φ is a S -invariant extended solution with uniton number 2:Φ λ = λ − π W + π ⊥ W ⊕ W + λπ W , (21) where W is a holomorphic isotropic subbundle of S × R n of rank l satisfying the superhorizontality condition ∂W ⊆ W ⊥ .Set ˜ V l = V l ⊕ V l and ˜ W = W ⊕ W . The T σ ̺,l -invariance of Φ implies that[ π W , π V ⊕ ˜ V l ] = 0 . (22)Now, write ϕ = g · V and consider the identification (20). We must haveΦ − = exp( πζ l ) gσ ̺ ( g − ) = exp( πζ l )( π ϕ − π ⊥ ϕ ) s . (23)From (21) and (23) we obtain π ϕ − π ⊥ ϕ = Ad( s ) (cid:0) π V ⊕ ˜ V l π ⊥ ˜ W + π ⊥ V ⊕ ˜ V l π ˜ W − π V ⊕ ˜ V l π ˜ W − π ⊥ V ⊕ ˜ V l π ⊥ ˜ W (cid:1) . (24)In view of (22), we see that π V ⊕ ˜ V l π ⊥ ˜ W + π ⊥ V ⊕ ˜ V l π ˜ W is an orthogonal projection, and (24) implies that this mustbe an orthogonal projection onto a 1-dimensional real subspace. Then, one of its two terms vanishes, that iseither ˜ W ⊂ V ⊕ ˜ V l or ˜ W ⊥ ⊂ ( V ⊕ ˜ V l ) ⊥ . For dimensional reasons, we see that the second case can not occur.Hence, we have π ϕ = Ad( s )( π V ⊕ ˜ V l π ⊥ ˜ W ) = π V ⊕ ˜ V l Ad( s )( π ⊥ ˜ W ) , that is (19) holds with R = V ⊕ V l ⊕ V l and A = s ( W ). (cid:3) Remark 4. If ϕ is full in R , then the isotropic subbundle A is the l -osculating space of some full totallyisotropic holomorphic map f from S into the complex projective space of R , the so called directrix curve of ϕ . That is, in a local system of coordinates ( U, z ), we have A ( z ) = Span (cid:8) g, g ′ , . . . , g ( l − } , where g is a lift of f over U and g ( r ) the r -th derivative of g with respect to z . Hence, formula (19) agrees with the classificationgiven in Corollary 6.11 of [9]. Example 1. Let us consider the case n = 2. We have only one class of harmonic maps: ( ζ , σ ̺, ). FromTheorem 20, any such harmonic map ϕ : S → R P is given by ϕ = R ∩ ( A ⊕ A ) ⊥ , where R is a constant3-dimensional subspace of R and A a holomorphic isotropic subbundle of S × R of rank 1 such that ∂A ⊆ A ⊥ .Taking into account Theorem 17, any such holomorphic subbundles A can be obtained from a meromorphicfunction a on S as follows.We have ζ = E and the corresponding extended solutions have uniton number r ( ζ ) = 1 (with respect tothe standard representation). Any extended solution Φ : S \ D → U σ ̺, ζ ( SO (4)) is given by Φ = exp C · γ ζ ,with γ ζ ( λ ) = λ − π V + π ⊥ V ⊕ V + λπ V , for some holomorphic vector-valued function C : S \ D → ( u ζ ) σ ̺, ,where ( u ζ ) σ ̺, = ( p ζ ) ⊥ ∩ k C σ ̺, = g ζ ∩ k C σ ̺, = ( g L − L ⊕ g L + L ) ∩ k C σ ̺, . Considering the root vectors X i,j , Y i,j , Z i,j as defined in (17), we have Y , = σ ̺, ( X , ). Hence C = a ( z )( X , + Y , ) where a ( z ) is a meromorphic function on S . In this case, from (2), it follows that (exp C ) − (exp C ) z = C z , and it is clear that the extended solution condition for Φ holds independently of the choice of the meromor-phic function a ( z ). Then, with respect to the complex basis u = { u , u , u , u } ,exp C · γ ζ = a − a a − a 00 0 1 00 0 − a · γ ζ (25)and the subbundle A of R = Span { u , u , u + u } is given by A = exp C · V = span { ( a , a, − , a ) } , whichsatisfies ∂A ⊆ A ⊥ . Example 2. Any harmonic two-sphere into R P in the class ( ζ , σ ̺, ) takes values in some R P inside R P and so it is essentially of the form (25). Next we consider the Weierstrass representation of harmonic spheresinto R P in the class ( ζ , σ ̺, ), which are given by ϕ = R ∩ ( A ⊕ A ) ⊥ , where R is a constant 5-dimensionalsubspace of R and A a holomorphic isotropic subbundle of S × R of rank 2 such that ∂A ⊆ A ⊥ . We have ζ = E + E , then r ( ζ ) = 2. Any extended solution Φ : S \ D → U σ ̺, ζ ( SO (6)) is given by Φ = exp C · γ ζ ,with γ ζ ( λ ) = λ − π V + π ⊥ V ⊕ V + λπ V , for some holomorphic vector-valued function C : S \ D → ( u ζ ) σ ̺, ,where ( u ζ ) σ ̺, = (cid:16) ( g L − L ⊕ g L + L ) ∩ k C σ ̺, (cid:17) ⊕ (cid:16) ( g L − L ⊕ g L + L ) ∩ k C σ ̺, (cid:17) ⊕ g L + L . We have Y , = σ ̺, ( X , ) and Y , = σ ̺, ( X , ). Hence we can write C = a ( z )( X , + Y , ) + b ( z )( X , + Y , ) + c ( z ) Y , where a ( z ), b ( z ) and c ( z ) are meromorphic functions on S .Now, Φ = exp C · γ ζ is an extended solution if and only if, in the expression C z − (ad C ) C z , whichdoes not depend on λ , the component on g ζ = g L + L vanishes. Since Y , = [ Y , , X , ] = [ X , , Y , ]and [ X , , X , ] = [ Y , , Y , ] = 0, this holds if and only if c ′ = ba ′ − ab ′ , where prime denotes z -derivative.Since A = exp C · V , we can compute exp C in order to conclude that the holomorphic subbundle A of R = Span { u , u , u , u , u + u } is given by A = Span { ( a , ab + c, a, − , , a ) , ( ab − c, b , b, , − , b ) } . Harmonic maps into Real Grassmanians. Let ζ ′ be a ̺ -semi-canonical element of SO (2 n ) given by (18),for some disjoint subsets I and I of { , . . . , n − } . By Proposition 9, we know that P σ ̺ ζ ′ ∼ = R P n − if andonly if either I = ∅ or I = { n − } . More generally we have: Proposition 21. If I = { i > i > . . . > i r } and d = P rj =1 ( − j +1 i j , then P σ ̺ ζ ′ ∼ = G d +1 ( R n ). Proof. For ζ ′ of the form (18), set ζ ′ I = P i ∈ I ζ i . Clearly, exp πζ ′ = exp πζ ′ I , and, by Proposition 6, P σ ̺ ζ ′ is asymmetric space with involution τ = Ad(exp πζ ′ I ) ◦ σ ̺ = Ad( s exp πζ ′ I ) . We have ζ ′ I = r ( E + . . . + E i r ) + ( r − E i r +1 + . . . + E i r − ) + . . . + ( E i +1 + . . . + E i ) , and consequently, with the convention V i = V n and V i r +1 = { } ,exp πζ ′ I = r X j =0 ( − j π i j − i j +1 + r X j =0 ( − j π i j − i j +1 , where π i j − i j +1 is the orthogonal projection onto V i j ∩ V ⊥ i j +1 and π i j − i j +1 the orthogonal projection onto the corre-sponding conjugate space. Hence, the +1-eigenspace of s exp ζ ′ I has dimension 2 d +1, with d = P rj =1 ( − j +1 i j ,which means that P σ ̺ ζ ′ ∼ = G d +1 ( R n ). (cid:3) In particular, we have P σ ̺ ζ d ∼ = G d +1 ( R n ) for each d ∈ { , . . . , n − } . Theorem 22. Each harmonic map from S into the real Grassmannian G d +1 ( R n ) belongs to one of thefollowing classes: ( ζ, Ad exp π (˜ ζ − ζ ) ◦ σ ̺,l ) , where ζ and ˜ ζ are ̺ -canonical elements such that ˜ ζ (cid:22) ζ and˜ ζ = P i ∈ I ζ i + ζ l , wherea) I = { i > i > . . . > i r } satisfies d = P rj =1 ( − j +1 i j ;b) l ∈ { , , . . . , n − } and l / ∈ I (if l = 0, we set ζ = 0). Proof. We consider harmonic maps into P σ ̺ ζ d ∼ = G d +1 ( R n ). Let ζ ′ be a ̺ -semi-canonical element and write ζ ′ = P i ∈ I ζ i + P i ∈ I ζ i for some disjoint subsets I and I of { , . . . , n − } . By Proposition 21, P σ ̺ ζ ′ ∼ = G d +1 ( R n ) if and only if either d = P rj =1 ( − j +1 i j or n − d − P rj =1 ( − j +1 i j , since G d +1 ( R n ) and G d ′ +1 ( R n ) , with d ′ = n − d − 1, can be identified via V V ⊥ . However, it follows from the same reasoningas in the proof of Theorem 19 that, in the second case, P σ ̺ ζ ′ does not coincide with the connected component P σ ̺ ζ d . So we only consider the ̺ -semi-canonical elements ζ ′ with d = P rj =1 ( − j +1 i j .Set l = max I . Next we check that the pair ζ ′ (cid:22) ˜ ζ = P i ∈ I ζ i + ζ l satisfies the conditions of Proposition 16.Considering the same notations we used in the proof of Theorem 19, for each i > ′ ̺ ∩ ∆ iζ ′ = { L j − L n | | I ∩ { j, . . . , l }| + | I ∩ { j, . . . , n − }| = 2 i } . In particular, for i > α = L j − L n ∈ ∆ ′ ̺ ∩ ∆ iζ ′ , it is clear that α ( ζ ′ − ˜ ζ ) / i ≤ i − , and consequently g ζ ′ i ∩ m C ̺ ⊂ M ≤ j< i g ζ ′ − ˜ ζj . For i > 0, we have the decomposition g ζ ′ i − ∩ k C ̺ = M α ∈ ∆( k ̺ ) ∩ ∆ i − ζ ′ g α ⊕ M α ∈ ∆ ′ ̺ ∩ ∆ i − ζ ′ ( g α ⊕ g ̺ ( α ) ) ∩ k C ̺ . Given α ∈ g ζ ′ i − , since α ( ζ ′ ) / i is odd, we must have α ( ζ j ) = 0 for some j ∈ I . Hence α ( ζ ′ − ˜ ζ ) / i < α ( ζ ′ ) / i andwe conclude that g ζ ′ i − ∩ k C ̺ ⊂ M ≤ j< i − g ζ ′ − ζj . The statement of the theorem follows now from Propositions 15 and 16. (cid:3) Next we will study in detail the case G ( R ). Take as base point of G ( R ) the 3-dimensional real subspace V ⊕ V ⊕ V , where V is the one-dimensional isotropic subspace spanned by u . This choice establishes theidentification G ( R ) ∼ = SO (6) /S ( O (3) × O (3))and the corresponding involution is σ ̺, = Ad(exp πζ ) ◦ σ ̺ . Following our classification procedure, we alsoidentify G ( R ) with P σ ̺ ζ via the totally geodesic embedding (13). From Theorem 22, we have six classes ofharmonic maps into G ( R ):( ζ , σ ̺ ) , ( ζ + ζ , σ ̺ ) , ( ζ , σ ̺, ) , ( ζ , σ ̺, ) , ( ζ + ζ , σ ̺, ) , ( ζ , Ad(exp πζ ) ◦ σ ̺, ) . Theorem 23. Let ϕ : S → G ( R ) be an harmonic map.(1) If ϕ is associated to the pair ( ζ , σ ̺ ) then ϕ is S -invariant and, up to isometry, is given by ϕ = V ⊕ V ⊕ V , (26)where V is a holomorphic isotropic subbundle of S × V ⊥ of rank 1 satisfying ∂V ⊆ V ⊥ .(2) If ϕ is associated to the pair ( ζ + ζ , σ ̺ ) and is S -invariant, then, up to isometry, ϕ = V ⊕ ( W ∩ V ⊥ ) ⊕ ( W ∩ V ⊥ ) , (27)where V ⊂ W are holomorphic isotropic subbundles of S × V ⊥ of rank 1 and 2, respectively, satisfying ∂V ⊂ W and ∂W ⊂ W ⊥ . (3) If ϕ is associated to the pair ( ζ , σ ̺, ) and is S -invariant, then, up to isometry, ϕ = { ( L ⊕ L ) ⊥ ∩ ( V ⊕ V ⊕ V ) } ⊕ ( L ⊕ L ) , (28)where L and L are holomorphic isotropic bundle lines of S × ( V ⊕ V ⊕ V ) and S × ( V ⊕ V ⊕ V ) ⊥ ,respectively. The corresponding extended solutions have uniton number 2, 4, and 2, respectively, with respect to the stan-dard representation of SO (6). The harmonic maps in the classes ( ζ , σ ̺, ), ( ζ + ζ , σ ̺, ), and ( ζ , Ad(exp πζ ) ◦ σ ̺, ) are precisely the orthogonal complements of the harmonic maps in the classes ( ζ , σ ̺ ), ( ζ + ζ , σ ̺ ), and( ζ , σ ̺, ), respectively. Proof. For the first two classes, and according to our classification procedure, we identify G ( R ) with P σ ̺ ζ via the totally geodesic embedding g · ( V ⊕ V ⊕ V ) exp( πζ ) gσ ̺, ( g − ) . In these two cases, T σ ̺ -invariantextended solutions Φ associated to harmonic maps ϕ = g · ( V ⊕ V ⊕ V ) satisfyΦ − = exp( πζ ) gσ ̺, ( g − ) = exp( πζ )( π ϕ − π ⊥ ϕ ) exp( πζ ) s . (29)First we consider the harmonic maps associated to the pair ( ζ , σ ̺ ). We have r ( ζ ) = 1 and( u ζ ) σ ̺ = ( p ζ ) ⊥ ∩ k C ̺ = g ζ ∩ k C ̺ . Consequently any such harmonic map is S -invariant. Write γ ζ ( λ ) = λ − π V + π ⊥ V ⊕ V + λπ V , where V isthe one-dimensional isotropic space spanned by u . Let Φ : S \ D → U σ ̺ ζ be an extended solution associatedto the harmonic map ϕ . Then, by S -invariance, we can writeΦ λ = λ − π V + π ⊥ V ⊕ V + λπ V , (30)where V is a holomorphic isotropic subbundle of S × R of rank 1 satisfying ∂V ⊆ V ⊥ . The T σ ̺ -invariance ofΦ implies that V ⊂ ( V ⊕ V ) ⊥ . Equating (29) and (30), we get, up to isometry, ϕ = V ⊕ V ⊕ V .For the case ( ζ + ζ , σ ̺ ), since γ ζ + ζ ( λ ) = λ − π V + λ − π V ∩ V ⊥ + π ⊥ V ⊕ V + λπ V ∩ V ⊥ + λ π V , (31)any S -invariant harmonic map ϕ in this class admits an extended solution of the formΦ λ = λ − π V + λ − π W ∩ V ⊥ + π ⊥ W ⊕ W + λπ W ∩ V ⊥ + λ π V , (32)where V ⊂ W are holomorphic isotropic subbundles of rank 1 and 2, respectively, satisfying ∂V ⊂ W and ∂W ⊂ W ⊥ . By T σ ̺ -invariance, we must have V ⊂ ( W ⊕ W ) ⊥ , hence V ⊂ W are subbundles of S × V ⊥ .Equating (29) and (32), we get (27).For the case ( ζ , σ ̺, ), we identify G ( R ) with P σ ̺, ζ = exp πζ P σ ̺ ζ − ζ via the totally geodesic embedding g · ( V ⊕ V ⊕ V ) gσ ̺, ( g − ) . (33)Extended solutions Φ associated to S -invariant harmonic maps in this class must be of the formΦ λ = λ − π W + π ⊥ W ⊕ W + λπ W , (34)where W is a holomorphic isotropic subbundle of rank 2. By T σ ̺, -invariance, we must have [ π W , π V ⊕ V ⊕ V ] = 0 , which means that W must be of the form W = L ⊕ L , where L and L , respectively, are holomorphic isotropicbundle lines of S × ( V ⊕ V ⊕ V ) and S × ( V ⊕ V ⊕ V ) ⊥ .On the other hand, in view of (33), we have Φ − = ( π ϕ − π ⊥ ϕ ) exp( πζ ) s . Equating this with (34), weconclude that (28) holds. The remaining cases are treated similarly. (cid:3) Remark 5. The first two classes of S -invariant harmonic maps ϕ : S → G ( R ) in Theorem 23 factor through G ( R ). That is, for any such harmonic map ϕ , there exists ˜ ϕ : S → G ( R ), where we identify R with V ⊥ ,such that ϕ = V ⊕ ˜ ϕ . An explicit construction of all harmonic maps from S into G ( R n ) can be found in [16].In that paper, harmonic maps of the form (26) are called real mixed pairs . We emphasise that the harmonicmaps into G ( R ) associated to extended solutions in the corresponding unstable manifolds need not to factorthrough G ( R ) in the same way. Let us consider the case ( ζ + ζ , σ ̺ ). Taking into account the Weierstrass representation of Theorem 17,any extended solution Φ : S \ D → U σ ̺ ζ ( SO (6)), with ζ = ζ + ζ , can be written as Φ = exp C · γ ζ , for somemeromorphic vector-valued function C : S → ( u ζ ) σ ̺ . We have r ( ζ ) = 3 and( u ζ ) σ ̺ = ( g ζ ⊕ g ζ ⊕ g ζ ) ∩ k C ̺ ⊕ λ ( g ζ ⊕ g ζ ) ∩ m C ̺ ⊕ λ g ζ ∩ k C ̺ . Moreover, g ζ ∩ k C ̺ = g L − L ⊕ { ( g L − L ⊕ g L + L ) ∩ k C ̺ } , g ζ ∩ k C ̺ = ( g L + L ⊕ g L − L ) ∩ k C ̺ , g ζ ∩ k C ̺ = g L + L , ( g ζ ⊕ g ζ ) ∩ m C ̺ = g ζ ∩ m C ̺ = ( g L − L ⊕ g L + L ) ∩ m C ̺ . Write C = C + λC + λ C , C = c + c + c , C = c + c , C = c (35)where the functions c i : S → g ζi ∩ k C ̺ , c i : S → g ζi ∩ m C ̺ , and c : S → g ζ ∩ k C ̺ are meromorphic functions.Clearly, c = 0. Consider the root vectors defined by (17). Since σ ̺ ( X , ) = − Y , and σ ̺ ( X , ) = − Y , , wecan write c = aX , + b ( X , − Y , ) , c = c ( X , − Y , ) , c = dY , , c = e ( X , + Y , ) , c = f X , in terms of C -valued meromorphic functions a , b , c , d , e , f .Taking into account the results of Section 3.2, Φ = exp C · γ ζ is an extended solution if and only if, in theexpression (exp C ) − (exp C ) z = C z − 12! (ad C ) C z + 13! (ad C ) C z , we have:a) the independent coefficient should have zero component in each g ζ and g ζ , that is c z − 12 [ c , c z ] = 0 , c z − 12 [ c , c z ] − 12 [ c , c z ] + 16 [ c , [ c , c z ]] = 0; (36)b) the λ coefficient should have zero component in g ζ , that is[ c , c z ] + [ c , c z ] = 0 . (37)From equations (36) we get the equations (prime denotes z -derivative)2 c ′ = ab ′ − ba ′ , d ′ = 3 cb ′ − bc ′ ; (38)on the other hand, observe that (37) always holds since[ c , c z ] + [ c , c z ] ∈ [ g ζ ∩ k C ̺ , g ζ ∩ m C ̺ ] ⊂ g ζ ∩ m C ̺ = { } . Hence we conclude that, any extended solution Φ : S \ D → U σ ̺ ζ ( SO (6)), with ζ = ζ + ζ , of the formΦ = exp C · γ ζ , can be constructed as follows: choose arbitrary meromorphic functions a , b , e and f ; integrateequations (38) to obtain the meromorphic functions c and d ; C is then given by (35). Example 3. Choose a ( z ) = b ( z ) = z . From (38), we can take c ( z ) = 1 and d ( z ) = z . This data defines thematrix C and the S -invariant extended solution Φ = exp C · γ ζ , where the loop γ ζ , with ζ = ζ + ζ , is givenby (31). The extended solutions Φ : S → U σ ̺ ζ ( SO (6)) satisfying Φ = u ζ ◦ Φ are of the form Φ = exp C · γ ζ ,where the matrix C = C + C λ + C λ is given by C = z z − 10 0 z − z − z z 00 0 0 0 0 00 0 0 − z − − z + eλ f λ − eλ − f λ eλ − eλ , with respect to the complex orthonormal basis u = { u , u , u , u , u , u } , where e and f are arbitrary mero-morphic functions on S . The holomorphic vector bundles V and W associated to the S -invariant extendedsolution exp C · γ ζ are given by V = exp C · V and W = exp C · V , and we have, with respect to the basis u , V = span { (12 − z − z , − z , − z , , − z, − 12 + 6 z ) } W = span { (6 z + z , z , z, , , − z ) } ⊕ V. Outer symmetric SU (2 n + 1) -spaces. Let E j be the square ( m × m )-matrix whose ( j, j )-entry is i andall other entries are 0. The complexification t C of the algebra t of diagonal matrices P a i E i , with a i ∈ C and P a i = 0, is a Cartan subalgebra of su ( m ) C . Let { L , . . . , L m } be the dual basis of { E , . . . , E m } , that is L i ( E j ) = i δ ij . The roots of su ( m ) are the vectors L i − L j , with i = j and 1 ≤ i, j ≤ m − + = { L i − L j } i 1. For i = j , the matrix X i,j whose ( i, j ) entry is 1 and all other entries are 0 generate the root space g L i − L j . The dual basis of∆ = { α , . . . , α m − } in i t ∗ is formed by the matrices H i = m − im ( E + . . . + E i ) − im ( E i +1 + . . . + E m ) . Special Lagrangian spaces. Consider on R m the standard inner product h· , ·i and the canonical orthonor-mal basis e m = { e , . . . , e m } . Define the orthogonal complex structure I by I ( e i ) = e m +1 − i , for i ∈ { , . . . , m } .A Lagrangian subspace of R m (with respect to I ) is a m -dimensional subspace L such that IL ⊥ L . Let L m be the space of all Lagrangian subspaces of R m and L ∈ L m the Lagrangian subspace generated by e m = { e , . . . , e m } . The unitary group U ( m ) acts transitively on L m , with isotropy group at L equal to SO ( m ), and L m is a reducible symmetric space that can be identified with U ( m ) /SO ( m ) (see [18] for details).The space L m can also be interpreted as the set of all orthogonal linear maps τ : R m → R m satisfying τ = e and Iτ = − τ I . Indeed, if V ± are the ± τ , then IV + = V − and IV + ⊥ V + , that is V + is Lagrangian. From this point of view, U ( m ) acts on L m by conjugation: g · τ = gτ g − . Let τ ∈ L m be theorthogonal linear map corresponding to L , that is, τ | L = e and τ | IL = − e . The corresponding involutionon U ( m ) is given by σ ( g ) = τ gτ and the Cartan embedding ι : L m ֒ → U ( m ) is given by ι ( τ ) = τ τ .The totally geodesic submanifold L sm := SU ( m ) /SO ( m ) of U ( m ) /SO ( m ) is also known as the space of specialLagrangian subspaces of R m . It is an irreducible outer symmetric SU ( m )-space.5.2.2. Harmonic maps into L s n +1 . Take m = 2 n + 1. The non-trivial involution ̺ of the Dynkin diagram of su (2 n + 1) C is given by ̺ ( α i ) = α n +1 − i . In particular, ̺ does not fix any root in ∆ and there exists only oneclass of outer symmetric SU (2 n + 1)-spaces. The semi-fundamental basis π k ̺ (∆ ) = { β , . . . , β n } is given by β i = ( α i + α n +1 − i ) whereas the dual basis { ζ , . . . , ζ n } is given by ζ i = H i + H n +1 − i = E + . . . + E i − ( E n +2 − i + . . . + E n +1 ) , for 1 ≤ i ≤ n . Since each ζ i belongs to the integer lattice I ( SU (2 n + 1)), the ̺ -semi-canonical elements of SU (2 n + 1) are precisely the elements ζ = P ni =1 m i ζ i with m i ∈ { , , } .Let e n +1 = { e , . . . , e n +1 } be the canonical orthonormal basis of R n +1 . Identify C n +1 with ( R n +2 , I ),where I is defined as above. Set v j = 1 √ e j + i e n +2 − j ) , for 1 ≤ j ≤ n , v n +1 = e n +1 and v n +2 − j = v j . Take the matrices E j with respect to the complex basis v = { v , . . . , v n +1 } of C n +1 . Hence τ E j τ = − E n +2 − j and the fundamental involution σ ̺ is given by σ ̺ ( g ) = τ gτ . The fundamental outer symmetric SU (2 n + 1)-space is the space of special Lagrangian subspaces L s n +1 = SU (2 n + 1) /SO (2 n + 1), and this is the unique outer symmetric SU (2 n + 1)-space.Next we consider in detail harmonic maps into L s . In this case we have two non-zero ̺ -semi-canonicalelements, ζ and 2 ζ , and consequently two classes of harmonic maps, ( ζ , σ ̺ ) and ( ζ , σ ̺, ). Since ζ = E − E , we have r ( ζ ) = ( L − L )( ζ ) / i = 2. Let W , W and W be the complex one-dimensional images of E , E and E , respectively. Any extended solutionΦ : S \ D → U σ ̺ ζ ( SU (2 n + 1))is given by Φ = exp C · γ ζ , with γ ζ ( λ ) = λ − π W + π W + λπ W , for some holomorphic vector-valued function C : S \ D → ( u ζ ) σ ̺ , where ( u ζ ) σ ̺ = ( p ζ ) ⊥ ∩ k C ̺ + λ ( p ζ ) ⊥ ∩ m C ̺ and ( p ζ ) ⊥ ∩ k C ̺ = ( g L − L ⊕ g L − L ⊕ g L − L ) ∩ k C ̺ , ( p ζ ) ⊥ ∩ m C ̺ = g L − L ∩ m C ̺ . Let X i,j be the square matrix whose ( i, j ) entry is 1 and all the other entries are 0, with respect to thebasis v . The root space g L i − L j is spanned by X i,j . We have σ ̺ ( X , ) = − X , and σ ̺ ( X , ) = − X , (consequently, g L − L ⊂ m C ̺ ). Hence we can write C = C + C λ , with C = a ( X , − X , ) and C = bX , ,for some meromorphic functions a, b on S . The harmonicity equations do not impose any condition on thesemeromorphic functions, hence any harmonic map ϕ : S → L s in the class ( ζ , σ ̺ ) admits an extended solutionof the form Φ = exp a bλ − a · γ ζ = a ( − a + 2 bλ )0 1 − a · γ ζ , (39)and ϕ is recovered by setting ϕ = Φ − τ . Similarly, one can see that the class of harmonic maps in ( ζ , σ ̺, )admits an extended solution of the formΦ = a ( a + 2 bλ )0 1 a · γ ζ , (40)with no restrictions on the meromorphic functions a and b .H. Ma established (cf. Theorem 4.1 of [13]) that harmonic maps ϕ : S → L s are essentially of two types:1) ι σ ◦ ϕ is a Grassmannian solution obtained from a full harmonic map f : S → R P ⊂ C P , where ι σ is the Cartan embedding of L s in SU (3); 2) up to left multiplication by a constant, ι σ ◦ ϕ is of the form( π β − π ⊥ β )( π β − π ⊥ β ), where β is a Frenet pair associated to a full totally istotropic holomorphic map g : S → C P and β is a rank 1 holomorphic subbundle of G ′ ( g ) ⊥ , where G ′ ( g ) is the first Gauss bundle of g .Observe that if, in the second case, β coincides with g , then ι σ ◦ ϕ is a Grassmannian solution obtained fromthe full harmonic map f := G ′ ( g ) from S to R P , that is, ϕ is of type 1). Comparing this with our description,it is not difficult to see that harmonic maps of type 1) are S -invariant extended solutions (take b = 0 in (39)and (40)) and harmonic maps of type 2) are associated to extended solutions with values in the correspondingunstable manifolds (which corresponds to an arbitrary choice of b in (39) and (40)). H. Ma also establisheda purely algebraic explicit construction of such harmonic maps in terms of meromorphic data on S , which isconsistent with our results.5.3. Outer symmetric SU (2 n ) -spaces. With the same notations of Section 5.2, the non-trivial involution ̺ of the Dynkin diagram of su (2 n ) is given by ̺ ( α i ) = α n − i , and ̺ fixes the root α n . The semi-fundamentalbasis π k ̺ (∆ ) = { β , . . . , β n − } is given by β = α n and β i = ( α i + α n − i ) if i ≥ 2; whereas its dual basis { ζ , . . . , ζ n − } is given by ζ = H n = 12 ( E + . . . + E n ) − 12 ( E n +1 + . . . + E n ) ζ i = H i − + H n − i +1 = E + . . . + E i − − ( E n +2 − i + . . . + E n ) , for 2 ≤ i ≤ n − . By Theorem 7, there exist two conjugacy classes of outer involutions: the fundamental outer involution σ ̺ and σ ̺, . These outer involutions correspond to the symmetric spaces SU (2 n ) /Sp ( n ) and SU (2 n ) /SO (2 n ),respectively. Observe that ζ does not belong to the integer lattice I ′ ( SU (2 n ) σ ̺ ) since exp 2 πζ = − e .5.3.1. Harmonic maps into the space of special unitary quaternionic structures on C n . A unitary quaterninonicstructure on the standard hermitian space ( C n , h· , ·i ) is a conjugate linear map J : C n → C n satisfying J = − Id and h v, w i = h J w, J v i for all v, w ∈ C n . Consider as base point the quaternionic structure J o definedby J o ( e i ) = e n +1 − i for each 1 ≤ i ≤ n , where e n = { e , . . . , e n } is the canonical hermitian basis of C n .The unitary group U (2 n ) acts transitively on the space of unitary quaternionic structures on C n with isotropygroup at J o equal to Sp ( n ), and thus M = U (2 n ) /Sp ( n ). This is a reducible symmetric space with involution σ : U (2 n ) → U (2 n ) given by σ ( X ) = J o XJ − o , but the totally geodesic submanifold Q sn := SU (2 n ) /Sp ( n ) is anirreducible symmetric space, which we call the space of special unitary quaternionic structures on C n (see [18]for details). If we consider the matrices E i with respect to the complex basis v = { v , . . . , v n } defined by v j = 1 √ e j + i e n +1 − j ) , (41)for 1 ≤ j ≤ n , and v n +1 − j = v j , we see that J o E j J − o = − E n +1 − j , and consequently we have σ = σ ̺ .Next we consider with detail harmonic maps into Q s . Proposition 24. Each harmonic map ϕ : S → Q s belongs to one of the following classes: (2 ζ , σ ̺ ), and( ζ , σ ̺, ). Proof. We start by identifying Q s with P σ ̺ e .The ̺ -semi-canonical elements of SU (4) are precisely the elements2 ζ , ζ , ζ , ζ , ζ + ζ , ζ + 2 ζ , ζ + ζ , ζ + 2 ζ . By Proposition 9, all these elements correspond to the symmetric space Q s .We claim that exp πζ is not in the connected component P σ ̺ e = { gJ o g − J − o | g ∈ SU (4) } . In fact, exp( πζ ) J o = gJ o g − ∼ = gSp ( n ) for the unitary transformation g defined by g ( e ) = e , g ( e ) = e , g ( e ) = e and g ( e ) = − e . Since det g = 1 we conclude that exp πζ does not belong to P σ ̺ e . Similarly, onecan check that exp π (2 ζ + ζ ) is not in P σ ̺ e .Hence, since exp π ζ belongs to the centre of SU (4), any harmonic map ϕ : S → Q s ∼ = P σ ̺ e belongs to oneof the following classes: (2 ζ , σ ̺ ), ( ζ , σ ̺, ), and (2 ζ + ζ , σ ̺, ). It remains to check that, in view of Proposition16, harmonic maps in the class (2 ζ + ζ , σ ̺, ) can be normalized to harmonic maps in the class ( ζ , σ ̺, ).It is clear that 2 ζ + ζ (cid:22) ζ . On the other hand, for any positive root L i − L j ∈ ∆ + , with i < j , wehave ( L i − L j )(2 ζ ) / i ≤ ( L i − L j )(2 ζ + ζ ) / i , where the equality holds in just one case: ( L − L )(2 ζ ) =( L − L )(2 ζ + ζ ) = 2i . However, g L − L ⊂ k σ ̺, , which means that the conditions of Proposition 16 hold for ζ = 2 ζ + ζ and ζ ′ = ζ , and consequently harmonic maps in the class (2 ζ + ζ , σ ̺, ) can be normalized toharmonic maps in the class ( ζ , σ ̺, ). (cid:3) Following the same procedure as before, one can see that any harmonic map ϕ → Q s in the class (2 ζ , σ ̺ )admits an extended solution of the formΦ = c + aλ c c c − aλ · γ ζ , where c , c , c ∈ C are constants, a is a meromorphic function on S . The harmonic map is recovered by setting ϕ = Φ − J o . Reciprocally, given arbitrary complex constants c , c , c and a meromorphic function a : S → C , such Φ is an extended solution associated to some harmonic map in the class (2 ζ , σ ̺ ) (the harmonicity equationsdo not impose any restriction to a ).Similarly, any harmonic map ϕ → Q s in the class ( ζ , σ ̺, ) admits an extended solution of the formΦ = b a c a − b · γ ζ , where a , b and c are meromorphic functions satisfying c ′ = ba ′ − b ′ a . Since P σ ̺, ζ = exp( πζ ) P σ ̺ e , the harmonicmap is recovered by setting ϕ = exp( πζ )Φ − J o .5.3.2. Harmonic maps into L s n . The outer symmetric SU (2 n )-space that corresponds to the involution σ ̺, isthe space of special Lagrangian subspaces L s n ∼ = SU (2 n ) /SO (2 n ). Take as base point the Lagrangian space L o = Span { e , . . . , e n } of R n and let τ be the corresponding conjugation, so that the Cartan embedding of L s n into SU (2 n ) is given by τ = gτ o g − gτ g − τ ∈ P σ ̺, e . Lemma 25. For each ζ ∈ I ( SU (2 n ) σ ̺, ) we have exp πζ ∈ P σ ̺, e . Proof. Each ζ ∈ I ( SU (2 n ) σ ̺, ) can be written as ζ = P ni =1 n i ( E i − E n +1 − i ) . Hence, exp πζ = π V − π ⊥ V ,where V = L n i even Span { e i , e n +1 − i } . Define g ∈ SU (2 n ) as follows: if n i is even, then g ( e i ) = e i and g ( e n +1 − i ) = e n +1 − i ; if n i is odd, then g ( e i ) = i e i and g ( e n +1 − i ) = − i e n +1 − i . We have exp πζ = gτ g − τ ,that is exp πζ ∈ P σ ̺, e . (cid:3) Now, identify L s n with P σ ̺, e via its Cartan embedding. By Theorem 12, any harmonic map ϕ : S → P σ ̺, e admits an extended solution Φ : S \ D → U σ ̺, ζ ′ ( SU (2 n )), for some ζ ′ ∈ I ′ ( SU (2 n )) ∩ k σ ̺, and some discretesubset D . We can assume that ζ ′ is a ̺ -semi-canonical element. The corresponding S -invariant solution u ζ ◦ Φtakes values in Ω ξ ( SU (2 n ) σ ̺, ), with ξ ∈ I ′ ( SU (2 n ) σ ̺, ); and both Φ − and ( u ζ ◦ Φ) − take values in P σ ̺, ξ . Apriori, ξ can be different from ζ since σ ̺, is not a fundamental outer involution. However, by Lemma 25 wehave P σ ̺, ξ = P σ ̺, e = P σ ̺, ζ ′ . If ζ is a ̺ -canonical element such that ζ ′ (cid:22) ζ and U ζ ′ ,ζ ′ − ζ (Φ) is constant, then, taking into account Proposition15, there exists a T τ -invariant extended solution ˜Φ : S \ D → U τζ ( SU (2 n )) , where τ = Ad(exp π ( ζ ′ − ζ )) ◦ σ ̺, . (42)such that ˜Φ − take values in P τζ and ϕ is given up to isometry by ϕ = exp( ζ ′ − ζ ) ˜Φ − τ . (43)We conclude that, given a pair ( ζ, τ ), where ζ ∈ I ( SU (2 n ) σ ̺ ) is a ̺ -canonical element and τ is an outerinvolution of the form (42), any extended solution ˜Φ : S \ D → U τζ ( SU (2 n ))) gives rise via (43) to an harmonicmap ϕ from the two-sphere into L s n and, conversely, all harmonic two-spheres into L s n arise in this way.For L s , since exp π ζ belongs to the centre of SU (4), we have five classes of harmonic maps into L s :(2 ζ , σ ̺, ) , ( ζ , σ ̺, ) , (2 ζ + ζ , σ ̺, ) ( ζ , Ad exp πζ ◦ σ ̺, ) , (2 ζ + ζ , Ad exp πζ ◦ σ ̺, ) . Let us consider in detail the class ( ζ , σ ̺, ). Clearly r ( ζ ) = 2. Let W , W , W and W be the complex one-dimensional images of E , E , E and E , respectively. That is, W i = Span { v i } , where v i are defined by (41).Any extended solution Φ : S \ D → U σ ̺, ζ is given by Φ = exp C · γ ζ , with γ ζ ( λ ) = λ − π W + π W ⊕ W + λπ W ,for some holomorphic vector-valued function C : S \ D → ( u ζ ) σ ̺, , where( u ζ ) σ ̺, = ( p ζ ) ⊥ ∩ k C σ ̺, + λ ( p ζ ) ⊥ ∩ m C σ ̺, and ( p ζ ) ⊥ ∩ k C σ ̺, = ( g L − L ⊕ g L − L ⊕ g L − L ⊕ g L − L ) ∩ k C σ ̺, , ( p ζ ) ⊥ ∩ m C σ ̺, = g L − L ∩ m C σ ̺, = g L − L . We have σ ̺, ( X , ) = − X , and σ ̺, ( X , ) = X , . Hence we can write C = C + C λ , with C = a ( X , − X , ) + b ( X , + X , ) , C = cX , for some meromorphic functions a, b, c on S . The harmonicity equations impose that ab ′ − ba ′ = 0, whichmeans that b = αa for some constant α ∈ C . Hence given arbitrary meromorphic functions a, c on S and acomplex constant α , Φ = a αa cλ − αa a · γ ζ , is an extended solution associated to some harmonic map in the class ( ζ , σ ̺, ). Reciprocally, any harmonicmap in such class arises in this way. 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