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

Proceedings of the 16th OCU International Academic Symposium 2008 OCAMI Studies Volume 3 2009, pp.41 52 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, 2009. 2000 Mathematics Subject Classification. Primary 58E20, Secondary 53C07. Key words and phrases. harmonic maps, Grassmannians, vector bundles. 41

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. 2.1. 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,

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). 2.2. 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).

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.

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.

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.

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},

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

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]). 4.2. 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.

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

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. 4.3. 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

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), 43 62 [2] J.Eells and J.H.Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160 [3] A.Gambioli, Latent Quaternionic Geometry, Tokyo Journal of Mathematics 31 (2008), 203 223 [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), 87 116 [7] M.Takeuchi and S.Kobayashi, Minimal imbeddings of R-spaces, J. Differential Geometry. 2 (1968), 203 215 [8] T.Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380-385 [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 819-0395, JAPAN E-mail address: nagatomo@math.kyushu-u.ac.jp