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HARMONIC MORPHISMS FROM THE CLASSICAL COMPACT SEMISIMPLE LIE GROUPS 7 0 (VERSION 1.059) 0 2 n SIGMUNDURGUDMUNDSSONANDANNASAKOVICH a J Abstract. Inthispaper weintroduceanewmethodformanufacturinghar- 9 monicmorphismsfromsemi-Riemannianmanifolds. Thisisemployedtoyield 1 a variety of new examples from the compact Lie groups SO(n), SU(n) and Sp(n) equipped with their standard Riemannian metrics. We develop a du- ] ality principle and show how this can be used to construct the first known G examples of harmonic morphisms from the non-compact Lie groups SLn(R), D SU∗(2n), Sp(n,R),SO∗(2n), SO(p,q),SU(p,q)andSp(p,q)equippedwith theirstandarddualsemi-Riemannianmetrics. . h t a m [ 1. Introduction 3 The notion of a minimal submanifold of a given ambient space is of great v importance in differential geometry. Harmonic morphisms φ : (M,g) 4 → 7 (N,h)between semi-Riemannian manifoldsareusefultools fortheconstruc- 5 tion of such objects. They are solutions to over-determined non-linear sys- 1 tems of partial differential equations determined by the geometric data of 1 the manifolds involved. For this reason harmonic morphisms are difficult to 6 0 find and have no general existence theory, not even locally. / For the existence of harmonic morphisms φ : (M,g) (N,h) it is an h → t advantage that the target manifold N is a surface i.e. of dimension 2. In a m this case the problem is invariant under conformal changes of the metric on N2. Therefore, at least for local studies, the codomain can be taken to be : v the complex plane with its standard flat metric. For the general theory of i X harmonic morphisms between semi-Riemannian manifolds we refer to the r excellent book [2] and the regularly updated on-line bibliography [6]. a The Riemannian manifold Sol is one of the eight famous 3-dimensional homogeneous geometries and has the structure of a Lie group compatible with its Riemannian metric. Baird and Wood have shown in [3] that Sol does not allow any local harmonic morphisms with values in a surface. This fact has been the main motivation for the research leading to this paper. We introduce thenotion of an eigenfamily of complex valued functions on a given semi-Riemannian manifold (M,g). We show how such a family can be used to construct a variety of harmonic morphisms on open and dense subsets of M. 2000 Mathematics Subject Classification. 58E20,53C43,53C12. Key words and phrases. harmonicmorphisms,minimalsubmanifolds,Liegroups. 1 We focus our attention on the classical semi-simple Lie groups and construct eigenfamilies in the compact Riemannian cases of SO(n), SU(n) and Sp(n) inducing a variety of new harmonic morphisms on these important spaces. The examples constructed by our new method are the first harmonic morphisms on SO(n), SU(n) and Sp(n) which are not invariant under the action of the subgroups SO(p) SO(q), S(U(p) U(q)) and Sp(p) Sp(q), × × × respectively, with n = p+q. For the known invariant solutions leading to harmonic morphisms on the Grassmannians, see [8]. In Theorem 7.1 we prove a usefulduality principle and show how this can be used to obtain eigenfamilies on the non-compact semi-Riemannian Lie groups SL (R), SU∗(2n), Sp(n,R), n ∗ SO (2n), SO(p,q), SU(p,q) and Sp(p,q). This leads to the construction of the first known examples of harmonic morphisms in all these cases. It should be noted that the non-compact semi-simple Lie groups SO(n,C), SL (C) and Sp(n,C) n arecomplex manifoldsandhencetheircoordinatefunctionsformorthogonal harmonic families, see Definition 2.4. This means that in these cases the problem is more or less trivial. Throughout this article we assume, when not stating otherwise, that all ∞ our objects such as manifolds, maps etc. are smooth, i.e. in the C - category. For our notation concerning Lie groups we refer to the wonderful book [10]. 2. Harmonic Morphisms LetM andN betwomanifoldsofdimensionsmandn,respectively. Then a semi-Riemannian metric g on M gives rise to the notion of a Laplacian on (M,g) and real-valued harmonic functions f : (M,g) R. This can → be generalized to the concept of a harmonic map φ : (M,g) (N,h) be- → tween semi-Riemannian manifolds being a solution to a semi-linear system of partial differential equations, see [2]. Definition 2.1. A map φ : (M,g) (N,h) between semi-Riemannian → manifolds is called a harmonic morphism if, for any harmonic function f : U R defined on an open subset U of N with φ−1(U) non-empty, the com→position f φ: φ−1(U) R is a harmonic function. ◦ → The following characterization of harmonic morphisms between semi- Riemannian manifolds is due to Fuglede and generalizes the corresponding 2 well-known result of [4, 9] in the Riemannian case. For the definition of horizontal conformality we refer to [2]. Theorem 2.2. [5] A map φ : (M,g) (N,h) between semi-Riemannian → manifolds is a harmonic morphism if and only if it is a horizontally (weakly) conformal harmonic map. The next result generalizes the corresponding well-known theorem of Baird and Eells in the Riemannian case, see [1]. It gives the theory of harmonic morphisms a strong geometric flavour and shows that the case when n = 2 is particularly interesting. In that case the conditions charac- terizing harmonic morphisms are independent of conformal changes of the metric on the surface N2. For the definition of horizontal homothety we refer to [2]. Theorem 2.3. [7] Letφ :(M,g) (Nn,h) be a horizontally conformal sub- → mersion from a semi-Riemannian manifold (M,g) to a Riemannian manifold (N,h). If i. n = 2 then φ is harmonic if and only if φ has minimal fibres, ii. n 3 then two of the following conditions imply the other: ≥ (a) φ is a harmonic map, (b) φ has minimal fibres, (c) φ is horizontally homothetic. Inwhatfollowswearemainlyinterestedincomplexvaluedfunctionsφ,ψ : (M,g) C from semi-Riemannian manifolds. In this situation the metric → g induces the complex-valued Laplacian τ(φ) and the gradient grad(φ) with C valuesinthecomplexified tangentbundleT M ofM. Weextendthemetric C g tobecomplex bilinearonT M anddefinethesymmetricbilinearoperator κ by κ(φ,ψ) = g(grad(φ),grad(ψ)). Two maps φ,ψ : M C are said to be orthogonal if → κ(φ,ψ) = 0. The harmonicity and horizontal conformality of φ : (M,g) C are given → by the following relations τ(φ) = 0 and κ(φ,φ) = 0. Definition 2.4. Let (M,g) be a semi-Riemannian manifold. Then a set = φ :M C i I i E { → | ∈ } of complex valued functions is said to be an eigenfamily on M if there exist complex numbers λ,µ C such that ∈ τ(φ) = λφ and κ(φ,ψ) = µφψ for all φ,ψ . A set ∈ E Ω = φ :M C i I i { → | ∈ } 3 is said to be an orthogonal harmonic family on M if for all φ,ψ Ω ∈ τ(φ) =0 and κ(φ,ψ) = 0. Thenextresultshowsthataneigenfamilyonasemi-Riemannianmanifold can be used to produce a variety of local harmonic morphisms. Theorem 2.5. Let (M,g) be a semi-Riemannian manifold and = φ ,...,φ 1 n E { } be a finite eigenfamily of complex valued functions on M. If P,Q : Cn C are linearily independent homogeneous polynomials of the same positi→ve degree then the quotient P(φ ,...,φ ) 1 n Q(φ ,...,φ ) 1 n is a non-constant harmonic morphism on the open and dense subset p M Q(φ (p),...,φ (p)) = 0 . 1 n { ∈ | 6 } A proof of Theorem 2.5 can be found in Appendix A. For orthogonal harmonic families we have the following useful result. Theorem 2.6. [7] Let (M,g) be a semi-Riemannian manifold and φ : M C k = 1,...,n k { → | } be a finite orthogonal harmonic family on (M,g). Let Φ : M Cn be the map given by Φ = (φ ,...,φ ) and U be an open subset of C→n containing 1 n the image Φ(M) of Φ. If = h :U C i I i H { → | ∈ } is a family of holomorphic functions then ψ : M C ψ = h(φ ,...,φ ), h 1 n { → | ∈ H} is an orthogonal harmonic family on M. 3. The Riemannian Lie group GL (C) n Let G be a Lie group with Lie algebra g of left-invariant vector fields on G. Then a Euclidean scalar product g on the algebra g induces a left- invariant Riemannianmetric on thegroupGandturnsitinto aRiemannian manifold. If Z is a left-invariant vector field on G and φ,ψ : U C are → two complex valued functions defined locally on G then the first and second order derivatives satisfy d Z(φ)(p) = [φ(p exp(sZ))] , ds · s=0 (cid:12) (cid:12) d2 Z2(φ)(p) = [φ(p exp(sZ))] . ds2 · s=0 (cid:12) 4 (cid:12) The tension field τ(φ) and the κ-operator κ(φ,ψ) are given by τ(φ) = Z2(φ) and κ(φ,ψ) = Z(φ)Z(ψ) ZX∈B ZX∈B where is any orthonormal basis of the Lie algebra g. Let GBL (C) be the complex general linear group equipped with its stan- n dardRiemannianmetric inducedby theEuclidean scalar producton theLie algebra gl (C) given by n ∗ g(Z,W) = RetraceZW . For 1 i,j n we shall by E denote the element of gl (R) satisfying ij n ≤ ≤ (E ) = δ δ ij kl ik jl and by D the diagonal matrices t D = E . t tt For 1 r < s n let X and Y be the matrices satisfying rs rs ≤ ≤ 1 1 X = (E +E ), Y = (E E ). rs rs sr rs rs sr √2 √2 − With the above notation we have the following easily verified matrix identities n (n 1) (n 1) X2 = − I , Y2 = − I , D2 = I , rs 2 n rs − 2 n t n Xr<s Xr<s Xt=1 1 X E Xt = (E +δ (I 2E )), rs jl rs 2 lj lj n − lj Xr<s 1 Y E Yt = (E δ I ), rs jl rs −2 lj − lj n Xr<s n D E Dt = δ E . t jl t jl lj Xt=1 4. The Riemannian Lie group SO(n) In this section we construct eigenfamilies of complex valued functions on the special orthogonal group SO(n) = x GL (R) x xt = I , detx= 1 . n n { ∈ | · } The Lie algebra so(n) of SO(n) is the set of skew-symmetric matrices so(n)= X gl (R) X +Xt = 0 n { ∈ | } and for this we have the canonical orthonormal basis Y 1 r < s n . rs { | ≤ ≤ } 5 Lemma 4.1. For 1 i,j n let x : SO(n) R be the real valued ij ≤ ≤ → coordinate functions given by x :x e x et ij i j 7→ · · where e ,...,e is the canonical basis for Rn. Then the following relations 1 n { } hold (n 1) τ(x )= − x , ij ij − 2 n 1 κ(x ,x ) = (x x δ x x ). ij kl il kj jl it kt −2 − Xt=1 Proof. It follows directly from the definition of the functions x that if X is ij anelementoftheLiealgebraso(n)thenthefirstandsecondorderderivatives satisfy X(x ): x e x X et and X2(x ): x e x X2 et. ij i j ij i j 7→ · · · 7→ · · · Employing the above mentioned matrix identities we then yield (n 1) τ(x )= Y2(x ) = e x Y2 et = − x , ij rs ij i· · rs· j − 2 ij Xr<s Xr<s κ(x ,x ) = e x Y et e Yt xt et ij kl i rs j l rs k · · · · · · · Xr<s = e x Y E Yt xt et i rs jl rs k · · · · · · (cid:0)Xr<s (cid:1) n 1 = (x x δ x x ). il kj jl it kt −2 − Xt=1 (cid:3) Let P,Q : SO(n) C be homogeneous polynomials of the coordinate functions x : SO(n)→ C of degree one i.e. of the form ij → n n P = trace(A xt)= a x and Q = trace(B xt)= b x ij ij kl kl · · iX,j=1 kX,l=1 for some A,B Cn×n. As a direct consequence of Lemma 4.1 we then yield ∈ PQ+2κ(P,Q) n n = a b x x +2 a b κ(x ,x ) ij kl ij kl ij kl ij kl i,jX,k,l=1 i,jX,k,l=1 n n n = a b x x a b x x + a b x x ij kl ij kl ij kl kj il ij kj it kt − i,jX,k,l=1 i,jX,k,l=1 i,jX,k,t=1 n = (a b a b )x x +trace(ABtxxt). ij kl kj il ij kl − i,jX,k,l=1 6 Comparing coefficients we see that PQ+2κ(P,Q) = 0 if ABt = 0 and a b ij il det = (a b a b ) = 0 (cid:18)akj bkl(cid:19) ij kl− kj il for all 1 i,j,k,l n. ≤ ≤ Theorem 4.2. Let V be a maximal isotropic subspace of Cn and p Cn be ∈ a non-zero element. Then the set (p)= φ :SO(n) C φ (x)= trace(ptaxt), a V V a a E { → | ∈ } of complex valued functions is an eigenfamily on SO(n). Proof. Assume that a,b V and define A = pta and B = ptb. By construc- ∈ tion any two columns of the matrices A and B are linearly dependent. This means that for all 1 i,j,k,l n ≤ ≤ a b det ij il = (a b a b )= 0. (cid:18)akj bkl(cid:19) ij kl− kj il FurthermorewehaveABt = 0. HenceP2+2κ(P,P) = 0,PQ+2κ(P,Q) = 0, Q2+2κ(Q,Q) = 0 and the statement follows directly from Lemma 4.1 and the calculations above. (cid:3) Applying the fact that xxt = I for each x SO(n) we get a simplified ∈ formula for the κ operator 1 κ(x ,x )= (δ δ x x ). ij kl ik jl il kj 2 − With a similar analysis to that above one yields the result of Theorem 4.3. It should be noted that having employed the special property xxt = I the eigenfamily (p) can not be used directly in the duality of Theorem 7.1. E Theorem 4.3. Let p be a non-zero isotropic element of Cn i.e. such that (p,p)= 0. Then the set (p)= φ :SO(n) C φ (x) = trace(ptaxt), a Cn a a E { → | ∈ } of complex valued functions is an eigenfamily on SO(n). Example4.4. Forz,w Cletpbetheisotropicelementofthe4-dimensional complex vector space C4∈given by p(z,w) = (1+zw,i(1 zw),i(z +w),z w). − − Thisgivesusthecomplex2-dimensionaldeformationofeigenfamilies each p E consisting of complex valued functions φ : SO(4) C a → of the form φ (x) = (1+zw)(a x +a x +a x +a x ) a 1 11 2 21 3 31 4 41 +i(1 zw)(a x +a x +a x +a x ) 1 12 2 22 3 32 4 42 − +i(z+w)(a x +a x +a x +a x ) 1 13 2 23 3 33 4 43 7 +(z w)(a x +a x +a x +a x ). 1 14 2 24 3 34 4 44 − 5. The Riemannian Lie group SU(n) In this section we construct eigenfamilies of complex valued functions on the unitary group U(n). They can be used to construct local harmonic morphisms on the special unitary group SU(n). The unitary group U(n) is the compact subgroup of GL (C) given by n U(n) = z GL (C) z z∗ =I . n n { ∈ | · } The circle group S1 = w C w = 1 acts on the unitary group U(n) by { ∈ | | | } multiplication (eiθ,z) eiθz and the orbit space of this action is the special 7→ unitary group SU(n)= z U(n) detz =1 . { ∈ | } The Lie algebra u(n) of the unitary group U(n) satisfies u(n)= Z Cn×n Z +Z∗ = 0 { ∈ | } and for this we have the canonical orthonormal basis Y ,iX 1 r < s n iD t = 1,...,n . rs rs t { | ≤ ≤ }∪{ | } Lemma 5.1. For 1 i,j n let z : U(n) C be the complex valued ij ≤ ≤ → coordinate functions given by z : z e z et ij i j 7→ · · where e ,...,e is the canonical basis for Cn. Then the following relations 1 n { } hold τ(z )= nz and κ(z ,z ) = z z . ij ij ij kl il kj − − Proof. The proof is similar to that of Lemma 4.1. (cid:3) Let P,Q : U(n) C be homogeneous polynomials of the coordinate functions z :U(n) →C of degree one i.e. of the form ij → n n P = trace(A zt) = a z and Q = trace(B zt) = b z ij ij kl kl · · iX,j=1 kX,l=1 for some A,B Cn×n. As a direct consequence of Lemma 5.1 we then yield ∈ n PQ+κ(P,Q) = (a b a b )z z . ij kl kj il ij kl − i,jX,k,l=1 Comparingcoefficientsweseethatκ(P,Q)+PQ = 0ifforall1 i,j,k,l n ≤ ≤ a b det ij il = (a b a b )= 0. (cid:18)akj bkl(cid:19) ij kl− kj il Theorem 5.2. Let p be a non-zero element of Cn. Then the set (p) = φ :U(n) C φ (z) = trace(ptazt), a Cn a a E { → | ∈ } of complex valued functions is an eigenfamily on U(n). 8 Proof. Assume that a,b Cn and define A = pta and B = ptb. By con- ∈ struction any two columns of the matrices A and B are linearly dependent. This means that for all 1 i,j,k,l n ≤ ≤ a b ij il det = (a b a b ) = 0 (cid:18)akj bkl(cid:19) ij kl− kj il soP2+κ(P,P) = 0,PQ+κ(P,Q) = 0andQ2+κ(Q,Q) = 0. Thestatement now follows directly from Lemma 5.1. (cid:3) It should be noted that the local harmonic morphisms on the unitary group U(n) that we obtain by employing Theorem 2.5 are invariant under the cirle action and hence induce local harmonic morphisms on the special unitary group SU(n). Example 5.3. The 3-dimensional sphere S3 is diffeomorphic to the special unitary group SU(2) given by z w SU(2) = z 2 + w 2 = 1 . {(cid:18) w¯ z¯(cid:19)| | | | | } − For p = (1,0) C2 we get the eigenfamily ∈ (p) = φ : U(n) C φ (z) = a z+a w, a= (a ,a ) Cn . a a 1 2 1 2 E { → | ∈ } By choosing a = (1,0) and b = (0,1) and applying Theorem 2.5 we obtain the well known globally defined harmonic morphism φ φ = a :SU(2) S2 φ → b called the Hopf map satisfying z w z φ( ) = . (cid:18) w¯ z¯(cid:19) w − 6. The Riemannian Lie group Sp(n) Inthissectionweconstructeigenfamiliesofcomplexvaluedfunctionsfrom the quaternionic unitary group Sp(n) being the intersection of the unitary group U(2n) and the standard representation of the quaternionic general linear group GL (H) in C2n×2n given by n z w (z+jw) q = . 7→ (cid:18) w¯ z¯(cid:19) − The Lie algebra sp(n) of Sp(n) satisfies sp(n) = Z W C2n×2n Z∗+Z =0, Wt W = 0 {(cid:18) W¯ Z¯(cid:19) ∈ | − } − and for this we have the standard orthonormal basis which is the union of the following three sets 1 Y 0 1 iX 0 rs rs , 1 r < s n , {√2(cid:18) 0 Yrs(cid:19) √2 (cid:18) 0 iXrs(cid:19) | ≤ ≤ } − 9 1 0 iX 1 0 X rs rs , 1 r < s n , {√2 (cid:18)iXrs 0 (cid:19) √2 (cid:18) Xrs 0 (cid:19) | ≤ ≤ } − 1 iD 0 1 0 iD 1 0 D t t t , , 1 t n . {√2(cid:18) 0 iDt(cid:19) √2 (cid:18)iDt 0 (cid:19) √2(cid:18) Dt 0 (cid:19) | ≤ ≤ } − − Lemma 6.1. For 1 i,j n let z ,w :Sp(n) C be the complex valued ij ij ≤ ≤ → coordinate functions given by z : g e g et, w : g e g et ij 7→ i· · j ij 7→ i· · n+j where e ,...,e is the canonical basis for C2n. Then the following rela- 1 2n { } tions hold 2n+1 2n+1 τ(z )= z , τ(w ) = w , ij ij ij ij − 2 · − 2 · 1 1 κ(z ,z ) = z z , κ(w ,w )= w w , ij kl il kj ij kl il kj −2 · −2 · n 1 κ(z ,w )= w z δ (z w w z ) . ij kl il kj jl it kt it kt −2 − · − (cid:2) Xt=1 (cid:3) Proof. The proof is similar to that of Lemma 4.1 but more involved. (cid:3) Theorem 6.2. Let p be a non-zero element of Cn. Then the set (p) = φ :Sp(n) C φ (g) = trace(ptazt +ptbwt), a,b Cn ab ab E { → | ∈ } of complex valued functions is an eigenfamily on Sp(n). Proof. Let a,b,c,d be arbitrary elements of Cn and define the complex valued functions P,Q :Sp(n) C by → P = trace(ptazt+ptbwt) and Q = trace(ptczt+ptdwt). Then a simple calculation shows that n PQ+2κ(P,Q) = [(a,d) (b,c)] (z w w z ) = 0. it kt it kt − − i,kX,t=1 Automatically we also get P2+2κ(P,P) = 0 and Q2+2κ(Q,Q) = 0. (cid:3) 7. The Duality In this section we show how a real analytic eigenfamily on a semi- E Riemannian non-compact semi-simple Lie group G gives rise to a real- ∗ analytic eigenfamily on its Riemannian compact dual U and vice versa. E The method of proof is borrowed from a related duality principle for harmonic morphisms from Riemannian symmetric spaces, see [8]. Let W be an open subset of G and φ : W C be a real analytic map. C → Let G denote the complexification of the Lie group G. Then φ extends uniquely to a holomorphic map φC : WC C from some open subset WC → 10