HARMONIC MAPS INTO GRASSMANNIANS AND A GENERALIZATION OF DO CARMO-WALLACH THEOREM

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1 Proceedings of the 16th OCU International Academic Symposium 2008 OCAMI Studies Volume , pp HARMONIC MAPS INTO GRASSMANNIANS AND A GENERALIZATION OF DO CARMO-WALLACH THEOREM YASUYUKI NAGATOMO Abstract. A harmonic map from a Riemannian manifold into a Grassmannian manifold is characterized by a vector bundle, a space of sections of the bundle and a Laplace operator. We apply the main result to generalize a Theorem of do Carmo and Wallach (Ann. of Math. 93 (1971) [1]) and describe a moduli space of harmonic maps with constant energy densities and some properties about pull-back bundle and connection from an isotropy irreducible compact reductive Riemannian homogeneous space into a Grassmannian. 1. Introduction One of the main theorems in this note concerns with a generalization of the Theorem of Takahashi [8], which we now review. Let f : M S N 1 be a smooth map from a Riemannian manifold (M, g) into a standard sphere S N 1 considered as a unit sphere of a Euclidean space R N. Denote the standard co-ordinates of R N by (x 1,, x N ). Then each co-ordinate function x A (A = 1,, N) can be regarded as a function on S N 1 by restriction. We can pull-back each x A by f : M S N 1 to obtain a function on M, which is also denoted by the same symbol. Then (a version of) Theorem of Takahashi asserts Theorem 1.1 ([8]). A map f : M S N 1 is a harmonic map if and only if there exsits a function h : M R such that x A = hx A for all A = 1,, N, where is the Laplace operator of (M, g). Under these conditions, we have h = df 2. Our concern is with a map from (M, g) into a real or complex Grassmannian manifold Gr p (W ) with a standard metric of Fubini-Study type, where W is a real or complex vector space with a scalar product. Let S Gr p (W ) be a tautological bundle. Since S Gr p (W ) is a subbundle of a trivial bundle W = Gr p (W ) W Gr p (W ), we have a quotient bundle Q Gr p (W ), which is called the universal quotient bundle. The scalar product on W gives an identification Q Gr p (W ) with an orthogonal complement of S Gr p (W ) in W Gr p (W ). Consequently, vector bundles S, Q Gr p (W ) are equipped with fibre metrics and connections. When a standard sphere S N 1 is identified with a real Grassmannian of oriented (N 1)-planes in R N, the universal quotient bundle Q Gr N 1 (R N ) is also the normal bundle. Then R N induces sections of the bundle and we recover functions x 1,, x N Date: Received on September 29, Mathematics Subject Classification. Primary 58E20, Secondary 53C07. Key words and phrases. harmonic maps, Grassmannians, vector bundles. 41

2 42 YASUYUKI NAGATOMO as sections. Moreover, since the bundle has a prefered connection, the Laplace operator on the bundle is well-defined. Hence we can reformulate Theorem of Takahashi from the viewpoint of vector bundles and a space of sections of the bundle, when these geometric structures are pull-backed. Then we have Main Theorem. Let (M, g) be an n-dimensional Riemannian manifold and f : M Gr p (W ) a smooth map. We fix a scalar product on W, which gives a Riemannian structure on Gr p (W ). Then, the following two conditions are equivalent. (1) f : M Gr p (W ) is a harmonic map. (2) W has the zero property for the Laplacian. Under these conditions, we have for an arbitrary t W, t = At, and df 2 = trace A, where the vector space W is regarded as a space of sections of the pull-back bundle f Q M. See Definition 2 for the zero property and a bundle homomorphism A is defined as the trace of the composition of the second fundamental forms in 3. We apply Main Theorem to obtain a generalization of Theorem of do Carmo and Wallach [1] in 4. In Theorem of do Carmo and Wallach, they apply Theorem of Takahashi to classify minimal immersions of spheres into spheres. A minimal immersion is a special case of harmonic maps with constant energy densities. A harmonic map with which we are concerned is a map from an isotropy irreducible compact reductive Riemannian homogeneous space G/K, say into a Grassmannian manifold. We take an irreducible homogeneous vector bundle with a canonical connection over G/K. Then each eigenspace of the Laplace operator on the vector bundle induces a G-equivariant map from G/K into a Grassmannian manifold, which is called a standard map. We give a sufficient condition for a standard map being harmonic (Lemma 4.4). Since a standard map is G- equivariant, the energy density is constant, which can be expressed with an eigenvalue. A few examples of standard maps are exhibited, some of which are related to Kähler or quaternion-kähler moment maps. We will use our Main Theorem to obtain a classification of harmonic maps with constant energy densities which have some properties about the pull-back bundle and connection (Theorem 4.6). Such a harmonic map is obtained as a deformation of a standard map. More details on the results described in this note can be found in [5]. 2. Preliminaries We review some standard material, mostly in order to fix our notation in this note A harmonic map. Let M and N be Riemannian manifolds and f : M N be a (differentiable) map. The energy density e(f) : M R of f is defined as e(f)(x) := 1 2 df 2 = 1 2 dim M i=1 df(e i ) 2,

3 HARMONIC MAP 43 where we use both of the Riemannian metrics on M and N and denote by e 1,, e dim M an orthonormal basis of the tangent space T x M. Then, the tension field τ(f) of f is defined to be τ(f) := trace df = dim M i=1 ( ei df)(e i ), which is a section of the pull-back bundle f T N M of the tangent bundle T N N. Definition 1. [2] A map f : M N is called a harmonic map if the tension field vanishes (τ(f) 0) geometry of Grassmannian manifolds. Let W be a real (oriented) or complex N-dimensional vector space and Gr p (W ) a Grassmannian manifold of (oriented) p-planes in W. The tautological vector bundle is denoted by S Gr p (W ). Since S Gr p (W ) is regarded as a subbundle of a trivial bundle W Gr p (W ) of a fibre W, we have an exact sequence of vector bundles: 0 S i S W π Q Q 0. The quotient vector bundle Q Gr p (W ) is called a universal quotient bundle. Then, the (holomorphic) tangent bundle T Gr p (W ) is identified with S Q. Note that the vector space W can be considered as a subspace of sections of Q Gr p (W ). From now on, we fix a scalar product (an inner product or a Hermitian product) on W. It gives orthogonal projections and so, we obtain two bundle homomorphisms: π S : W S, i Q : Q W. Then the vector bundles S Gr p (W ) and Q Gr p (W ) are equipped with fibre metrics g S and g Q, respectively. Using a trivialization of W Gr p (W ) by an orthogonal basis, we regard a section of W Gr p (W ) as a W -valued function. Usual differentiation d yields connections on the vector bundles S Gr p (W ) and Q Gr p (W ), which are denoted by S and Q, respectively. More precisely, if s is a section of S Gr p (W ), then we regard s as a W -valued function i S (s). Then di S (s) can be decomposed into two components: di S (s) = π S di S (s) + π Q di S (s). Indeed, π S di S (s) is a connection and nothing but S s, which is so-called the canonical connection. The other term π Q di S (s) is called the second fundamental form in the sense of Kobayashi [4], which is a 1-form with values in Hom(S, Q) = S Q. We put H := π Q di S. In a similar way, the connection Q is explicitly written down and we can define the second fundamental form K := π S di Q, which is a 1-form with values in Hom(Q, S) = Q S. The Levi-Civita connection is the same as the induced connection from S and Q. Lemma 2.1. The second fundamental forms H and K are parallel. Lemma 2.2. The second fundamental forms H and K satisfy g Q (Hs, t) = g S (s, Kt).

4 44 YASUYUKI NAGATOMO The main difference of a complex Grassmannian from a real Grassmannian is that we can use the Hodge decomposition, because a complex Grassmannian is a Kähler manifold. More precisely, let W be a complex vector space with a Hermitian product (, ) W and Gr p (W ) a complex Grassmannian of p-planes in W. We can define homogeneous vector bundles S Gr p (W ) and Q Gr p (W ) with induced Hermitian metrics g S and g Q by W, respectively. Canonical connections give holomorphic structures to S Gr p (W ) and Q Gr p (W ). In particular, W can be regarded as the space of holomorphic sections of Q Gr p (W ). The holomorphic tangent bundle T is identified with S Q and the holomorphic cotangent bundle is S Q. The identification includes Hermitian metrics and connections. The second fundamental form H Ω 1 (Hom(S, Q)) is of type (1, 0) and K Ω 1 (Hom(Q, S)) is of type (0, 1). 3. Harmonic maps into Grassmannians If f : M Gr p (W ) is a smooth map, then we pull back a fiber metric and a connection on Q Gr p (W ) to obtain a fibre metric g V and a connection V on the pull-back bundle f Q M, which is denoted by V M. In a similar way, the pull-back bundle U M of S Gr p (W ) has a pull-back fibre metric g U and a pull-back connection U. The second fundamental forms are also pull-backed and denoted by the same symbols H Γ(f T U V ) and K Γ(f T V U). If we restrict bundle-valued linear forms H and K on the pull-back bundle f T M to linear forms on M, then H and K are nothing but the second fundamental forms of subbundles U W and V W, respectively, where W = M W. We can pull back an exact sequence of vector bundles: 0 U i U W π V V 0. Note that, in this case, we have only a linear map W Γ(f Q), because it may not be an injection. However, we shall say that W is a space of sections of V M, even if the linear map is not injective. From now on, we assume that M is a Riemannian manifold with a metric g. Then, we use the Riemannian structure on M and the pull-back connection on V M to define the Laplace operator V = = V V = n ( ) i=1 V e i V (e i ) acting on sections of V M and a bundle homomorphism A Γ (Hom V ) is defined as the trace of the composition of the second fundamental forms H and K: n A := H ei K ei, i=1 where n is the dimension of M and {e i } i=1,2, n is an orthonormal basis of the tangent space of M. A bundle homomorphism A Γ (Hom V ) is called the mean curvature operator of (V M, W ). We can easily show properties of A Γ (Hom V ). Lemma 3.1. The mean curvature operator A is a non-positive symmetric (or Hermitian) operator. Lemma 3.2. The energy density e(f) is equal to 1 2trace A.

5 HARMONIC MAP 45 Let t be a section of V M. We denote by Z t the zero set of t: Z t := {x M t(x) = 0}. Definition 2. A space of sections W of a vector bundle V M has the zero property for the Laplacian if Z t Z t for an arbitrary t W. Example. If W is an eigen-space for the Laplacian, then W has the zero property. Theorem 3.3. Let (M, g) be an n-dimensional Riemannian manifold and f : M Gr p (W ) a smooth map. We fix an inner product or a Hermitian product (, ) on W, which gives a Riemannian structure on Gr p (W ). Then, the following two conditions are equivalent. (1) f : M Gr p (W ) is a harmonic map. (2) W has the zero property for the Laplacian. Under these conditions, we have for an arbitrary t W, t = At, and 2e(f) = trace A. Proof. First of all, we have H 2 = df 2, and so δ H = τ(f)(= H τ(f) ), where τ(f) can be regarded as a section of f T = f (S Q) M. We fix a vector w W and take the corresponding sections s W Γ(U) and t W Γ(V ) to w. Then we have ( V t ) (Y ) = H Y K X t H ( X df)(y )s. In particular, we obtain V X t + ( δ H ) s + n H ei K ei t = t + ( δ H ) s + At = 0. (3.1) i=1 First, suppose the condition (1). Since f : M Gr p (W ) is harmonic, it follows that the equation (3.1) reduces to t + At = 0. (3.2) We immediately deduce that W has the zero property. Conversely, we suppose the condition (2). For an arbitrary vector u U x, x M, we can find an element w W such that the corresponding sections s Γ(U) and t Γ(V ) satisfy s(x) = u, and t(x) = 0. The equation (3.1) yields that H τ(f) s = t + At. Since W has the zero property for the Laplacian and t(x) = 0, it follows that t(x) = 0. Hence we have H τ(f) u = 0, and so τ(f) = 0, which means that f is a harmonic map.

6 46 YASUYUKI NAGATOMO Theorem 3.4. Let (M, g) be an n-dimensional Rimannian manifold and f : M Gr p (W ) a map. We fix an inner product or a Hermitian product (, ) on W. Suppose that V M is holonomy irreducible with respect to the induced connection. Then, the following two conditions are equivalent. (1) f : M Gr p (W ) is a harmonic map and there exists a function h(x) such that A x = h(x)id V for an arbitrary x M. (2) There exists a function h on M such that t = ht for an arbitrary t W. Moreover, under the above conditions, we have 2e(f) = qh, where e(f) is the energy density of f and q = rank Q. Proof. First, we suppose the condition (1). It immediately follows from the equation (3.2) and A = h(x)id V that t = ht. Conversely, we suppose the condition (2). It yields that W has the zero property, and so f is a harmonic map. It follows from the equation (3.1) that and so, we get t + At = 0, At = ht. The non-positivity of A as an operator yields that h is non-negative and we have trace A = qh. 4. A generalization of Theorem of do Carmo and Wallach In this section, we shall give a generalization of do Carmo-Wallach theorem. Definition 3. Let V M be a vector bundle and W a space of sections of V M. We define an evaluation homomorphism ev : W V in such a way that ev(t)(x) := t(x) V x for t W. The vector bundle V M is said to be globally generated by W if the evaluation homomorphism ev : W V is surjective. Definition 4. Let V M be a real vector bundle of rank q which is globally generated by W of dimension N. If V M is oriented, then we also fix an orientation on W. Then we have a map to a real (oriented) Grassmannian f : M Gr p (W ), where p = N q defined as f(x) := Ker ev x = {t W t(x) = 0}. A map f : M Gr p (W ) is called an induced map by (V M, W ), or simply W, if the vector bundle V M is specified. If V M is a complex vector bundle and W is a complex vector space of sections which globally generates V M, then we have a map to a complex Grassmannian in a similar method. It is also called the induced map.

7 HARMONIC MAP 47 Remark. Assume that W has a subspace W 0 which consists of only zero section. We take a complementary subspace W 1 of W 0 in W. Then W globally generates V M if and only if so does W 1. In addition, the induced map by W is the composition of the induced map by W 1 and a natural inclusion Gr p1 (W 1 ) Gr p (W ), where p 1 = dim W 1 q. To remove the case, V M is globally and effectively generated by W if only zero vector in W corresponds to the zero section. From our definition of an induced map f : M Gr p (W ), the vector bundle V M can be naturally identified with the pull back bundle f Q M. Conversely, if f : M Gr p (W ) is a smooth map, then W can be regarded as a space of sections of Q Gr p (W ) which is globally generated by W. Pulling back Q Gr p (W ) to M, we obtain a vector bundle V M which is also globally generated by W. It is easily observed that the induced map by W is the same as the original map f : M Gr p (W ). In this way, every map f : M Gr p (W ) can be recognized as an induced map. Let M = G/K be an isotropy irreducible compact reductive Riemannian homogeneous space with decomposition g = k m, where G is a compact Lie group and K is a closed subgroup of G. Let V 0 be a q-dimensional K-irreducible real or complex representation space with a K-invariant scalar product. We can construct a homogeneous vector bundle V M, V := G K V 0 with an invariant fibre metric g V induced by the scalar product on V 0. Moreover V M has a canonical connection with respect to the decomposition g = k m. (This means that the horizontal distribution is defined as {L g m g G} on a principal fibre bundle G M.) A Lie group G naturally acts on the space of sections Γ(V ) of V M, which has a G-invariant L 2 -inner product. Using the Levi-Civita connection and, we can decompose the space of sections of V M into the eigen-spaces of the Laplacian: Γ(V ) = µ W µ, W µ := {t Γ(V ) t = µt}. It is well-known that W µ is a finite dimensional G-representation space. Lemma 4.1. The vector bundle V M is globally generated by W µ, if µ 0. Proof. Frobenius reciprocity and the irreducibility of V 0 yield the result. Lemma 4.2. Under the same assumption as in Lemma 4.1, V 0 can be regarded as a subspace of W µ. Proof. If the action on W of G is restricted to a subgroup K, then the evaluation W µ V 0 = V [e] is K-equivariant, where [e] G/K and e is a unit element. Since W µ has a G-invariant L 2 -inner product, we can take the orthogonal complement of Ker ev [e] denoted by ( ). Ker ev [e] Schur s lemma implies the result and we get a K-representation V 0 = ( ). Ker ev [e] 4.1. Standard maps. We can define the induced map f 0 : M Gr p (W µ ) by W µ, where p = N q, N = dim W µ, which is called the standard map by W µ. f 0 ([g]) = {t W µ t([g]) = 0},

8 48 YASUYUKI NAGATOMO The orthogonal complement of V 0 W µ is denoted by U 0. Then we can express the induced map f 0 : M Gr p (W µ ) as f 0 ([g]) = gu 0 W µ, which is G-equivariant. The irreducibility and Schur s lemma show that we can assume that the induced metric on V M is just the original metric g V. Next, we consider the pull-back connection V. Then we have Lemma 4.3. The pull-back connection V is the cannonical connection if and only if mv 0 U 0. Lemma 4.4. If a G-module (ϱ, W µ ) satisfies the condition mv 0 U 0, then the standard map f 0 : M Gr p (W ) is harmonic and we have 2e(f 0 ) = qµ, A = µid V Proof. The hypothesis yields that V M is holonomy irreducible. We apply Theorem 3.4 to obtain the result. In general, W µ is not irreducible as G-representation. If we take an irreducible G- subspace W of W µ, then Lemma 4.1 is still valid. We can consider the induced map, which is also called the standard map by W. In a similar vein, we have Lemma 4.5. Let (ϱ, W ) be an irreducible G-submodule of (ϱ, W µ ). Then we can regard V 0 as a subspace of W and U 0 denotes the orthogonal complement of V 0 in W. If (ϱ, W ) satisfies the condition mv 0 U 0, then the standard map f 0 : M Gr p (W ) is harmonic and we have 2e(f 0 ) = qµ, A = µid V We take some examples from [5]. Example. Let CP 1 = SU(2)/U(1) be a complex projective line and O(1) CP 1 a holomorphic line bundle of degree 1 with a canonical connection. Frobenius reciprocity yields that the symmetric power S 2n+1 C 2 of the standard representation C 2 (n Z 0 ) is an SU(2)-invariant space of sections of O(1) CP 1, where (ϱ 2n+1, S 2n+1 C 2 ) is an irreducible representation of SU(2). Moreover, S 2n+1 C 2 is an eigen-space of the Laplacian (see [9]). We denote by C k (k Z) an irreducible U(1)-module with weight k. As a homogeneous vector bundle, O(1) CP 1 may be regarded as SU(2) U(1) C 1. We can regard C 1 as a weight subspace of S 2n+1 C 2 and this provides us with the standard evaluation S 2n+1 C 2 O(1). It follows that ϱ 2n+1 (m)c 1 C 3 C 1, because the complexification of m is identified with C 2 C 2. Consequently, the standard map f 0 : CP 1 CP 2n = P(S 2n+1 C 2 ) is a harmonic map from Lemma 4.5. See also [6] about an equivariant harmonic map into a complex projective space. Example. Let M = HP 1 = Sp(2)/Sp(1) Sp(1) be a quaternion projective space. To destinguish two copies of Sp(1) in the isotoropy subgroup, we write the isotropy subgroup as Sp + (1) Sp (1). Let H be the standard representation of Sp + (1) and E be the standard representation of Sp (1). Then the associated homogeneous vecotr bundles

9 HARMONIC MAP 49 are denoted by the same symbols H M and E M, respectively. We suppose that H M is the tautological vector bundle and E M is the orthogonal complement in a trivial bundle H 2 = C 4 M. We take the symmetric power S k H M of H M and S l E M of E M. When k (resp.l) is even, S k H (resp.s l E) has a real structure. If the both of k and l are odd, then S k H S l E has a real structure. In those cases, for example, S k H is supposed to represent a real representation or the associated real vector bundle. Since the Lie algebra sp(2) is decomposed as Sp + (1) Sp (1)-module into: sp(2) = S 2 H S 2 E (H E), m = H E and sp(2) can be regarded as an eigen-space of the Laplacian acting on sections of S 2 H M. Then we have [m, S 2 H] m, because (Sp(2), Sp(1) Sp(1)) is a symmetric pair. Lemma 4.5 implies that the standard map f 0 : HP 1 Gr 7 (sp(2)) = Gr 7 (R 10 ) is a harmonic map. Now the standard map has another interpretation (see also Gambioli [3]). Let µ : HP 1 sp(2) S 2 H be a quaternion moment map. By definitin of a moment map, for an arbitrary X sp(2), we have µ X ([g]) = [ g, π S2 H(g 1 Xg) ], g Sp(2), where π S2 H : sp(2) S 2 H is the orthgonal projection. It follows that sp(2) is a subspace of sections of S 2 H M by the moment map µ. It is clear that sp(2) globally generates S 2 H M. We can define the induced map f µ : HP 1 Gr 7 (R 10 ). By definition of the induced map, we have f µ ([g]) = { X sp(2) Ad(g 1 )X S 2 E (H E) } =Ad(g) ( S 2 E (H E) ) sp(2), which is the same as the standard map f 0. The standard map induced by S 2 H HP 1 and sp(2) can be generalized on any compact quaternion symmetric space. It is induced by a quaternion moment map for an isometry group in the same way. Example. Let G/K be a compact irreducible Hermitian symmetric space. We can consider a moment map µ : G/K g. In this situation, µ X : G/K R for an arbitrary X g is an eigenfunction of the Laplacian. Then the theorem of Takahashi [8] yields that the induced map f : G/K S g is a harmonic map, where S is a hypersphere of g (Takeuchi-Kobayashi [7]) A harmonic map with constant energy density. Let G be a compact Lie group and W be a real representation of G with an invariant inner product (, ) W. We denote by S 2 W the symmetric power of W and by (, ) S the induced inner product on S 2 W. The inner product on S 2 W is supposed to satisfy (B, u v) S = (Bu, v) W, for any symmetric transform B : W W and any u, v W. It is easily checked that (, ) S is G-invariant.

10 50 YASUYUKI NAGATOMO If W is a complex representation with a Hermitian product (, ) W, then H(W ) denotes the set of Hermitian endomorphisms of W. We equip H(W ) with an inner product (, ) H ; (A, B) H := trace AB, A, B H(W ). It is also easily checked that (, ) H is G-invariant. We define a Hermitian operator H(u, v) for u, v W as H(u, v) := 1 2 {u (, v) W + v (, u) W }. Then we have for an arbitrary B H(W ) that (B, H(u, v)) H = 1 2 {(Bu, v) W + (Bv, u) W }. If U and V are subspaces of W, we define a real subspace H(U, V ) H(W ) spanned by H(u, v) where u U and v V. In this section, K denotes R or C. A symmetric operator B S 2 W is also called a Hermitian operator, for simplicity. Definition 5. Let f : M Gr p (K m ) be a map and we regard K m as a space of sections of the pull-back bundle f Q M. Then the map f : M Gr p (K m ) is called a full map if f Q M is effectively generated by K m. We are now in a position to state the main theorem in this section. Theorem 4.6. We fix an irreducible homogeneous vector bundle V = G K V 0 M := G/K of rank r with a canonical connection. Let f : M Gr p (K m ) be a full harmonic map with a constant energy density satisfying the following three conditions. (i) The pull-back bundle f Q M is isomorphic to V M. (Hence, r = m p.) (ii) The pull-back connection on f Q M is the canonical connection. (iii) The mean curvature operator A Γ(V ) arising from a map f can be expressed as h(x)id V for some function h(x). Then we have (I) an eigen-space W Γ(V ) with an eigenvalue µ of the Laplacian which has K m as a linear subspace. As a G-representation space (ϱ, W ), ϱ(m)v 0 V 0, (4.1) where V 0 is considered as a subspace of W. (II) There exists a positive Hermitian transform T Aut (W ) such that T V 0 V 0 and ( C, G(S 2 V 0 ) ) S = 0, (C, G(ϱ(m)V 0 V 0 )) S = 0, (4.2) or (C, GH(V 0, V 0 )) H = 0, (C, GH(ϱ(m)V 0, V 0 )) H = 0 where C := T 2 Id W. We regard Gr p (K m ) as a submanifold of Gr p (W ) (N p = m p) by U U K m, where U is a p-plane of K m and K m is the orthogonal complement of K m. Then, a map f : M Gr p (W ) can be expressed as f ([g]) = T f 0 ([g]) = T gu 0, 2e(f) = (m p)µ, h(x) = µ. (4.3) Conversely, suppose that an eigen-space W with an eigenvalue µ satisfies the condition (4.1) as a representation space and a Hermitian transform C Aut (W ) satisfies the condition (4.2) and C + Id W is positive. We put T := C + Id W. If T V 0 V 0 and

11 HARMONIC MAP 51 we define a map f : M Gr p (W ) as in (4.3), then f : M Gr p (W ) is a harmonic map with a constant energy density satisfying the conditions (i) and (ii) and the mean curvature operator A Γ(V ) is µid V. For a proof of Theorem 4.6, see [5]. We apply Theorem 4.6 to the case that the domain is an irreducible Hermitian symmmetric space of compact type. Corollary 4.7. Let M := G/K be an irreducible Hermitian symmmetric space of compact type. We take an irreducible homogeneous complex vector bundle V = G K V 0 M of rank r with a canonical connection which gives a holomorphic structure on V M. Let f : M Gr p (C m ) be a full holomorphic map with a constant energy density satisfying the following two conditions. (i) The pull-back bundle f Q M is metrically isomorphic to V M. (Hence, r = m p.) (ii) The mean curvature operator A Γ(V ) arising from a map f can be expressed as h(x)id V for some function h(x). Then we have (I) the space of holomorphic sections W := H 0 (M; V ) is an eigen-space of the Laplacian with an eigenvalue µ and has C m as a linear subspace. (II) There exists a positive Hermitian transform T Aut (W ) such that T V 0 V 0 and (C, GH(V 0, V 0 )) H = 0, (4.4) where C := T 2 Id W. We regard Gr p (C m ) as a submanifold of Gr p (W ) (N p = m p) in a similar manner to Theorem 4.6. Then, a map f : M Gr p (W ) can be expressed as f ([g]) = T f 0 ([g]) = T gu 0, 2e(f) = (m p)µ, h(x) = µ. (4.5) Conversely, suppose that W = H 0 (M; V ) is non-trivial and a Hermitian transform C Aut (W ) satisfies the condition (4.4) and C + Id W is positive. We put T := C + IdW. If T V 0 V 0 and we define a map f : M Gr p (W ) as in (4.5), then f : M Gr p (W ) is a holomorphic map with a constant energy density satisfying the conditions (i) and (ii) and the mean curvature operator A Γ(V ) is µid V, where µ is the eigenvalue of the Laplacian. Proof. We state the main difference from a proof of Theorem 4.6. First of all, let U(1) be the center of the isotropy subgroup K. Then m with the complex structure has an only positive weight as U(1)-module. Since V G/K has non-trivial holomorphic section, the Bott-Borel-Weil theorem and the irreducibility imply that V 0 has an only positive weight as U(1)-module. Consequently, ϱ (m) V 0 and V 0 have no common weight. It follows that ϱ(m)v 0 V 0. The pull-back bundle f Q M is holomorphically isomorphic to V M, because f : G/K Gr p (C m ) is a holomorphic map. Combined with the condition (i), the uniqueness of the compatible connection yields that the pull-back connection is the canonical connection. Therefore we do not have to consider conditions about connections Comparison with the ADHM-construction. Let M denote the 4-dimensional sphere S 4 = HP 1. We follow notation of Example after Lemma 4.5 and review the

12 52 YASUYUKI NAGATOMO ADHM-construction of instantons. For simplicity, we focus our attention on 1-instantons. Let α : C 4 H be a surjective bundle homomorphism satisfying Dα = 0, where α is regarded as a section of C 4 H S 4 and D is the twistor operator. Suppose that C 4 has an invariant Hermitian product and an invariant quaternion structure j under the action of Sp(2). Then we have the induced real structure of C 4 H = C 4 H. Using the twistor space and the Bott-Borel-Weil theorem, α can be expressed as α [g] (w) = [ g, π(g 1 T 1 w) ], g Sp(2), where T is an automorphism of C 4. The ADHM-construction requires that T should satisfy T T = Id + C, C ( 2 0C 4 ) R. Here 2 0C 4 is the orthogonal complement to Cω in 2 C 4, where ω is an invariant symplectic form on C 4, which is an irreducible representation of Sp(2). Since 2 0C 4 has an invariant real structure induced by j, we can take a real representation ( 2 0C 4 ) R. Then Ker α C 4 is an instanton with the induced metric and connection from C 4. If we regard α as an evaluation homomorphism, then we obtain the induced map f : M Gr 2 (C 4 ): f ([g]) = T ge. When T is the identity or C = O, we recover the standard map f 0. In the case that C O, the pull-back connection on the pull-back f Q M is not gauge equivalent to the canonical connection on H M. References [1] M.P.do Carmo and N.R.Wallach, Minimal immersions of spheres into spheres, Ann.Math. 93 (1971), [2] J.Eells and J.H.Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), [3] A.Gambioli, Latent Quaternionic Geometry, Tokyo Journal of Mathematics 31 (2008), [4] S.Kobayashi, Differential Geometry of Complex Vector Bundles, Iwanami Shoten and Princeton University, Tokyo (1987) [5] Y.Nagatomo, Harmonic maps into Grassmannian manifolds, a preprint [6] Y.Ohnita, Homogeneous Harmonic Maps into Complex Projective Spaces, Tokyo Journal of Mathematics 13 (1990), [7] M.Takeuchi and S.Kobayashi, Minimal imbeddings of R-spaces, J. Differential Geometry. 2 (1968), [8] T.Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), [9] N.R.Wallach, Harmonic Analysis on Homogeneous Spaces, Pure and Applied Mathematics, Marcel Dekker, INC, New York (1973) Graduate School of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka , JAPAN address: nagatomo@math.kyushu-u.ac.jp