HARMONIC MAPS and BIHARMONIC MAPS on PRINCIPAL BUNDLES and WARPED PRODUCTS 1. Introduction Variational Problems Play Central

J. Korean Math. Soc. 55 (2018), No. 3, pp. 553–574 https://doi.org/10.4134/JKMS.j170251 pISSN: 0304-9914 / eISSN: 2234-3008

HARMONIC MAPS AND BIHARMONIC MAPS ON PRINCIPAL BUNDLES AND WARPED PRODUCTS

Hajime Urakawa

Abstract. In this paper, we study harmonic maps and biharmonic maps on the principal G-bundle in Kobayashi and Nomizu [22] and also the ∞ warped product P = M ×f F for a C (M) function f on M studied by Bishop and O’Neill [4], and Ejiri [11].

1. Introduction Variational problems play central roles in geometry; Harmonic map is one of important variational problems which is a critical point of the energy functional 1 R 2 E(ϕ) = 2 M |dϕ| vg for smooth maps ϕ of (M, g) into (N, h) (see [7,16,17,20, 32,44,45,47]). The Euler-Lagrange equations are given by the vanishing of the tension ﬁled τ(ϕ). In 1983, J. Eells and L. Lemaire [10] extended the notion of harmonic map to biharmonic map, which are, by deﬁnition, critical points of the bienergy functional Z 1 2 (1.1) E2(ϕ) = |τ(ϕ)| vg. 2 M

After G. Y. Jiang [19] studied the ﬁrst and second variation formulas of E2, extensive studies in this area have been done (for instance, see [6, 14, 15, 18, 28, 31, 40, 41], etc.). Notice that harmonic maps are always biharmonic by deﬁnition. B. Y. Chen raised ([8]) so called B. Y. Chen’s conjecture and later, R. Caddeo, S. Montaldo, P. Piu and C. Oniciuc raised ([2,3,6]) the generalized B. Y. Chen’s conjecture.

B. Y. Chen’s conjecture. Every biharmonic submanifold of the Euclidean space Rn must be harmonic (minimal).

Received April 17, 2017; Revised January 5, 2018; Accepted January 30, 2018. 2010 Mathematics Subject Classiﬁcation. Primary 58E20; Secondary 53C43. Key words and phrases. principal G bundle, Ricci curvature, harmonic map, biharmonic map, warped product. Supported by the Grant-in-Aid for the Scientiﬁc Research, (C) No. 25400154, Japan Society for the Promotion of Science.

c 2018 Korean Mathematical Society 553 554 H. URAKAWA

The generalized B. Y. Chen’s conjecture. Every biharmonic submanifold of a Riemannian manifold of non-positive curvature must be harmonic (minimal). For the generalized Chen’s conjecture, Ou and Tang gave ([39,40]) a counter example in a Riemannian manifold of negative curvature. For the Chen’s conjecture, affirmative answers were known for the case of surfaces in the three dimensional Euclidean space ([8]), and the case of hypersurfaces of the four dimensional Euclidean space ([9, 13]). K. Akutagawa and S. Maeta gave ([1]) showed a supporting evidence to the Chen’s conjecture: Any complete regu- lar biharmonic submanifold of the Euclidean space Rn is harmonic (minimal). The affirmative answers to the generalized B. Y. Chen’s conjecture were shown ([34–36]) under the L2-condition and completeness of (M, g). In this paper, we first treat with a principal G-bundle over a Riemannian manifold, and show the following two theorems: Theorem 3.2. Let π :(P, g) → (M, h) be a principal G-bundle over a Rie- mannian manifold (M, h) with non-positive Ricci curvature. Assume P is compact so that M is also compact. If the projection π is biharmonic, then it is harmonic. Theorem 4.1. Let π :(P, g) → (M, h) be a principal G-bundle over a Rie- mannian manifold with non-positive Ricci curvature. Assume that (P, g) is a non-compact complete Riemannian manifold, and the projection π has both finite energy E(π) < ∞ and finite bienergy E2(π) < ∞. If π is biharmonic, then it is harmonic. We give two comments on the above theorems: For the generalized B. Y. Chen’s conjecture, non-positivity of the sectional curvature of the ambient space of biharmonic submanifolds is necessary. However, it should be em- phasized that for the principal G-bundles, we need not the assumption of non- positivity of the sectional curvature. We only assume non-positivity of the Ricci curvature of the domain manifolds in the proofs of Theorems 3.2 and 4.1. Second, finiteness of the energy and bienergy is necessary in Theorem 4.1. Otherwise, one have the counter examples due to Loubeau and Ou (cf. Sect. Four, Examples 1 and 2 [29]). Next, we consider the warped products. For two Riemannian manifolds (M, h), (F, k) and a C∞ function f on M, f ∈ C∞(M), the warping function on M, let us consider the warped product (P, g) where π : P = M × F 3 (x, y) 7→ x ∈ M and g = π∗h+f 2 k. Let us consider the following two problems: Problem 1. When π :(P, g) → (M, h) is harmonic?

Problem 2. In the case (M, h) = (R, dt2), a line, can one choose f ∈ C∞(R) such that π :(P, g) → (M, h) is biharmonic but not harmonic? In this paper, we answer these two problems as follows. BIHARMONIC MAPS ON PRINCIPAL BUNDLES & WARPED PRODUCTS 555

Theorem 5.2. Let π :(P, g) → (M, h) be the warped product with a warping function f ∈ C∞(M). Then, the tension field τ(π) is given by grad f ∇ f (1.2) τ(π) = ` = ` , f f where ` = dim F . Therefore, π is harmonic if and only if f is constant. Theorem 6.2. For the warped product π :(P, g) → (M, h), the bitension field τ2(π) is given by h (1.3) τ2(π) = ∆(τ(π)) − ρ (τ(π)) − ` ∇ ∇ f τ(π), f where ∆ is the rough Laplacian and ∇ is the induced connection from the Levi- Civita connection ∇h of (M, h). Therefore, π is biharmonic if and only if h (1.4) ∆(τ(π)) − ρ (τ(π)) − ` ∇ ∇ f τ(π) = 0. f h h Pm h 0 0 Here, ρ is the Ricci transform ρ (u) := i=1 R (u, ei)ei, u ∈ TxM for an 0 m locally defined orthonormal field {ei}i=1 on (M, h). Theorem 7.1. (1) In the case (M, h) = (R, dt2), a line, the warped product π :(P, g) → (R, dt2) is biharmonic if and only if f ∈ C∞(R) satisfies the following ordinary equation: (1.5) f 000 f 2 + (` − 3) f 00 f 0 f + (−` + 2) f 03 = 0. (2) All the solutions f of (1.5) are given by Z t ` (1.6) f(t) = c exp a tanh a r + b dr , t0 2 where a, b, c > 0 are arbitrary constants. (3) In the case (M, h) = (R, dt2), a line, let f(t) be C∞ function defined by (1.6) with a 6= 0 and c > 0. Then, the warped product π :(P, g) → (M, h) is biharmonic but not harmonic. Acknowledgement. We would like to express our gratitude to the referee who pointed to improve Theorem 3.2 in this version.

2. Preliminaries 2.1. Harmonic maps and biharmonic maps We first prepare the materials for the first and second variational formulas for the bienergy functional and biharmonic maps. Let us recall the definition of a harmonic map ϕ :(M, g) → (N, h), of a compact Riemannian manifold (M, g) into another Riemannian manifold (N, h), which is an extremal of the energy functional defined by Z E(ϕ) = e(ϕ) vg, M 556 H. URAKAWA

1 2 where e(ϕ) := 2 |dϕ| is called the energy density of ϕ. That is, for any variation {ϕt} of ϕ with ϕ0 = ϕ, Z d (2.1) E(ϕt) = − h(τ(ϕ),V )vg = 0, dt t=0 M where V ∈ Γ(ϕ−1TN) is a variation vector field along ϕ which is given by d V (x) = dt t=0ϕt(x) ∈ Tϕ(x)N,(x ∈ M), and the tension field is given by Pm −1 m τ(ϕ) = i=1 B(ϕ)(ei, ei) ∈ Γ(ϕ TN), where {ei}i=1 is a locally defined orthonormal frame field on (M, g), and B(ϕ) is the second fundamental form of ϕ defined by

B(ϕ)(X,Y ) = (∇e dϕ)(X,Y )

= (∇e X dϕ)(Y )

= ∇X (dϕ(Y )) − dϕ(∇X Y )(2.2) for all vector fields X,Y ∈ X(M). Here, ∇, and ∇h, are Levi-Civita connections on TM, TN of (M, g), (N, h), respectively, and ∇, and ∇e are the induced ones on ϕ−1TN, and T ∗M ⊗ ϕ−1TN, respectively. By (2.1), ϕ is harmonic if and only if τ(ϕ) = 0. The second variation formula is given as follows. Assume that ϕ is harmonic. Then, 2 Z d (2.3) 2 E(ϕt) = h(J(V ),V )vg, dt t=0 M where J is an elliptic differential operator, called the Jacobi operator acting on Γ(ϕ−1TN) given by (2.4) J(V ) = ∆V − R(V ), ∗ where ∆V = ∇ ∇V = − Pm {∇ ∇ V − ∇ V } is the rough Laplacian i=1 ei ei ∇ei ei and R is a linear operator on Γ(ϕ−1TN) given by m X N R(V ) = R (V, dϕ(ei))dϕ(ei), i=1 N h h h and R is the curvature tensor of (N, h) given by R (U, V ) = ∇ U ∇ V − h h h ∇ V ∇ U − ∇ [U,V ] for U, V ∈ X(N) (see [22, 27, 28]). J. Eells and L. Lemaire [10] proposed polyharmonic (k-harmonic) maps and Jiang [19] studied the first and second variation formulas of biharmonic maps. Let us consider the bienergy functional defined by Z 1 2 (2.5) E2(ϕ) = |τ(ϕ)| vg, 2 M where |V |2 = h(V,V ), V ∈ Γ(ϕ−1TN). BIHARMONIC MAPS ON PRINCIPAL BUNDLES & WARPED PRODUCTS 557

The ﬁrst variation formula of the bienergy functional is given by Z d (2.6) E2(ϕt) = − h(τ2(ϕ),V )vg. dt t=0 M Here,

(2.7) τ2(ϕ) := J(τ(ϕ)) = ∆(τ(ϕ)) − R(τ(ϕ)), which is called the bitension ﬁeld of ϕ, and J is given in (2.4). A smooth map ϕ of (M, g) into (N, h) is said to be biharmonic if τ2(ϕ) = 0. By deﬁnition, every harmonic map is biharmonic. We say, for an immersion ϕ : (M, g) → (N, h) to be proper biharmonic if it is biharmonic but not harmonic (minimal) (see [12, 23–27, 30, 33, 37, 38, 42, 43, 48, 49]).

2.2. The principal G-bundle Recall several notions on principal G-bundles ([5, 21, 22]). A manifold P = P (M,G) is a principal fiber bundle over M with a compact Lie group G, where p = dim P , m = dim M, and k = dim G. By definition, a Lie group G acts on P by right hand side denoted by (G, P ) 3 (a, u) 7→ u · a ∈ P , and, for each ∗ point u ∈ P , the tangent space TuP admits a subspace Gu := {A u | A ∈ g}, the vertical subspace at u, and each A ∈ g defines the fundamental vector field A∗ ∈ X(P ) by

∗ d A u := u exp(t A) ∈ TuP. dt t=0 A Riemannian metric g on P is called adapted if it is invariant under all the ∗ right action of G, i.e., Ra g = g for all a ∈ G. An adapted Riemannian metric on P always exists because for every Riemannian metric g0 on P , deﬁne a new metric g on P by Z 0 gu(Xu,Yu) = g (Ra∗Xu,Ra∗Yu) dµ(a), G ∗ where dµ(a) is a bi-invariant Haar measure on G. Then, Ra g = g for all a ∈ G. Each tangent space TuP has the orthogonal direct decomposition of the tangent space TuP ,

(a) TuP = Gu ⊕ Hu, where the subspace Gu of Pu satisﬁes ∗ (b) Gu = {A u | A ∈ g}, and the subspace Hu of Pu satisﬁes that

(C) Hu·a = Ra∗Hu, a ∈ G, u ∈ P, where the subspace Hu of Pu is called horizontal subspace at u ∈ P with respect to g. 558 H. URAKAWA

p In the following, we fix a locally defined orthonormal frame field {ei}i=1 m corresponding (a), (b) in such a way that {ei}i=1 is a locally defined orthonor- ∗ k mal basis of the horizontal subspace Hu (u ∈ P ), and {ei = A m+i}i=1 is a locally defined orthonormal basis of the vertical subspace Gu (u ∈ P ) for an k orthonormal basis {Am+i}i=1 of the Lie algebra g of a Lie group G with respect to the Ad(G) invariant inner product h·, ·i. For each decomposition (a), one can define a g-valued 1-form ω on P by V H ω(Xu) = A, Xu = Xu + Xu , where V H V ∗ Xu ∈ Gu,Xu ∈ Hu,Xu = Au for u ∈ P and a unique A ∈ g. This 1-form ω on P is called a connection form of P . Then, there exist a unique Riemannian metric h on M and an Ad(G)- invariant inner product h·, ·i on g such that

g(Xu,Yu) = h(π∗Xu, π∗Yu) + hω(Xu), ω(Yu)i,Xu,Yu ∈ TuP, u ∈ P, namely, g = π∗h + h ω(·), ω(·)i. We call this Riemannian metric g on P , an adapted Riemannian metric on P . Then, let us recall the following definitions for our question: Definition 2.1. (1) The projection π :(P, g) → (M, h) is to be harmonic if the tension field vanishes, τ(π) = 0, and (2) the projection π :(P, g) → (M, h) is to be biharmonic if, the bitension field vanishes, τ2(π) = J(τ(π)) = 0. Here, J is the Jacobi operator for the projection π given by J(V ) := ∆V − R(V ),V ∈ Γ(π−1TM), where p X ∆V := − ∇ (∇ V ) − ∇ V ei ei ∇ei ei i=1 m X = − ∇ (∇ V ) − ∇ V ei ei ∇ei ei i=1 k X ∗ ∗ ∗ − ∇A (∇A V ) − ∇∇A∗ A V m+i m+i m+i m+i i=1 −1 p for V ∈ Γ(π TM), i.e., V (x) ∈ Tπ(x)M (x ∈ P ). Here, {ei}i=1 is a local m orthonormal frame field on (P, g) which is given by that: {ei}i=1 is an orthonormal horizontal field on the principal G-bundle π :(P, g) → (M, h) and ∗ k {em+i, u = Am+i, u}i=1 (u ∈ P ) is an orthonormal frame field on the verti- ∗ cal space Gu = {Au| A ∈ g} (u ∈ P ) corresponding to an orthonormal basis k {Am+i}i=1 of (g, h, i). BIHARMONIC MAPS ON PRINCIPAL BUNDLES & WARPED PRODUCTS 559

2.3. The warped products On the product manifold P = M × F for two Riemannian manifolds (M, h) and (F, k), and a C∞ function, f ∈ C∞(M) on M, let us consider the Rie- mannian metric (2.8) g = π∗h + f 2 k, where the projection π : P = M × F 3 (x, y) 7→ x ∈ M. The Riemannian submersion π :(P, g) → (M, h) is called the warped product of (M, h) and (F, k) with a warping function f ∈ C∞(M) ([4, 11, 46]). In this section, we prepare several notions in order to calculate the tension field and bitension field. m+` We first construct a locally defined orthonormal frame field {ei}i=1 on (P, g) where m = dim M and ` = dim F as follows: For i = 1, . . . , m, 0 ei(x,y) := (e i x, 0y) ∈ T(x,y)P = TxM × TyF, and for i = m + 1, . . . , p, 1 e := (0 , e00 ) ∈ T P = T M × T F, i(x,y) f(x) x i, y (x,y) x y where p = m + `. Recall the O’Neill’s formulas on the warped product (cf. [4, 11]). For a C∞ vector field X ∈ X(M) on M, X∗ ∈ X(P ), the horizontal lift of X which satisfies for z ∈ P , ∗ ∗ (2.9) X z ∈ Hz, and π∗ (X z) = Xπ(z), where recall the vertical subspace Vz and horizontal subspace Hz of the tangent space TzP :

(2.10) Vz = Ker(π∗ (x,y)),

(2.11) TzP = Vz ⊕ Hz, g(Vz, Hz) = 0, where π∗ (x,y) : T(x,y)P → TxM is the differential of the projection π : P → M at (x, y) ∈ P . Let q : P = M × F 3 (x, y) 7→ y ∈ F be the projection of P onto F . For a vector field V on F , there exists a unique vector field Ve on P satisfying that Ve ∈ V and q∗(Ve) = V . We identify V ∈ X(F ) with Ve ∈ V denoting by the same letter V in the following. Lemma 2.1. Let X,Y ∈ X(M) be vector fields on M, and V,W ∈ X(F ), vector fields on F , and ∇g, ∇h, ∇k, the Levi-Civita connections of (P, g), (M, h), and (F, k), respectively. Then, (1) grad (f ◦ π) = grad f. g ∗ h ∗ ∗ (2) π∗ (∇ X∗ Y ) = ∇ X Y, where X and Y are the horizontal lifts of X and Y , respectively. g g ∗ Xf (3) ∇ X∗ V = ∇ V X = f V. 560 H. URAKAWA

g 1 (4) H (∇ V W ) = −f k(V,W ) G = − f g(V,W ) G, where G is the gradient of f and f ◦ π. g k (5) V (∇ V W ) = ∇ V W, where HA, and VA are the horizontal part, and the vertical part of A, respectively. Lemma 2.2 (O’Neill’s formulas). ∗ ∗ (1) g(x,y)(X (x,y),Y (x,y)) = hx(Xx,Yx), x ∈ M. ∗ ∗ (2) π∗ ([X ,Y ]) = [X,Y ]. g ∗ h (3) π∗ (∇ X∗ Y ) = ∇ X Y . g ∗ Lemma 2.3. For a vector field X ∈ X(M) whose h(X,X) is constant, ∇ X∗ X h is the horizontal lift of ∇ X X. g ∗ Proof. By (3) of Lemma 2.2, we only have to see ∇ X∗ X is a horizontal vector field. Due to Lemma 2.3(1), for every vertical vector field X ∈ X(M), we have g ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2g(∇ X∗ X ,V ) = X (g(X ,V )) + X (g(V,X )) − V (g(X ,X ) + g(V, [X∗,X∗]) + g(X∗, [V,X∗]) − g(X∗, [X∗,V ]) = 2g(X∗, [X∗,V ]) (2.12) = 0. Here, the last equality of (2.12) follows as: ∗ g g ∗ [X ,V ] = ∇ X∗ V − ∇ V X Xf Xf = V − V f f = 0(2.13) by using Lemma 2.1(3). Then, we can choose a locally defined orthonormal vector field

{e1, . . . , em, em+1, . . . , em+`} on (P, g) in such a way that {e1, . . . , em} are orthonormal vector ﬁelds which are 0 0 horizontal lifts of the orthonormal vector ﬁelds e1, . . . , em on (M, h) and em+1 = 1 00 1 00 g f em+1, . . . , em+` = f em+`. Then, by Lemma 2.3, ∇ ei ei, i = 1, . . . , m, are h 0 the horizontal lifts of ∇ 0 e . ei i For i = m + 1, . . . , m + `, we have the following decomposition:

g 1 n 00 k 00 o (2.14) ∇ ei ei = −(ei f) ei + ∇e00 ei − f ∇(f ◦ π) . f 2 i We first note that ∇(f ◦ π) is a horizontal vector field on P . Because, g(∇(f ◦ π),V ) = V f = 0 k 00 for every V ∈ X(F ). And the first two terms of (2.14) are vertical since ∇ 00 e , ei i i = m + 1, . . . , m + `, are vertical. BIHARMONIC MAPS ON PRINCIPAL BUNDLES & WARPED PRODUCTS 561

To prove (2.14), for i = m + 1, . . . , m + `, we have

g g 1 00 1 ∇ ei ei = ∇ e00 ei f i f 1 00 1 00 1 g 00 = e e + ∇ e00 e f i f i f i i 00 1 ei f 00 g 00 (2.15) = − e + ∇ e00 e . f 2 f i i i g 00 We decompose ∇ 00 e into the vertical and horizontal components: ei i g 00 g 00 g 00 (2.16) ∇ 00 e = V ∇ 00 e + H ∇ 00 e . ei i ei i ei i Here, by Lemma 2.1(5), we have g 00 k 00 (2.17) V ∇ 00 e = ∇ 00 e . ei i ei i 00 00 By Lemma 2.1(4) and k(ei , ej ) = δij, we have g 00 00 00 H ∇ 00 e = −f k(e , e ) G ei i i i = −f G = −f ∇ f = −f ∇ (f ◦ π)(2.18) by Lemma 2.1(1). We obtain (2.14).

3. Proof of Theorem 3.2 If the principal G-bundle π :(P, g) → (M, h) is harmonic, then it is clearly biharmonic. Our main interest is to ask the reverse holds under what condi- tions: Problem 3.1. If the projection π of a principal G-bundle π :(P, g) → (M, h) is biharmonic, is π harmonic or not. In this paper, we show that this problem is affirmative when the Ricci curvature of the base manifold (M, h) is negative definite. Indeed, we show that: Theorem 3.2. Let π :(P, g) → (M, h) be a principal G-bundle over a Rie- mannian manifold (M, h) with non-positive Ricci curvature. Assume P is compact so that M is also compact. If the projection π is biharmonic, then it is harmonic. In this section, we give a proof of Theorem 3.2 in case of a compact Riemann- ian manifold (M, h) and the Ricci tensor of (M, h) is negative definite. We will give the proof of Theorem 4.1 in case of a non-compact complete Riemannian manifold (M, h) in Section 4. Let us first consider a principal G-bundle π :(P, g) → (M, h) whose the total space P is compact. Assume that the projection π :(P, g) → (M, h) is 562 H. URAKAWA biharmonic, which is by definition, J(τ(π)) ≡ 0, where τ(π) is the tension field of π which is defined by p X (3.1) τ(π) := {∇h π e − π (∇ e )}, ei ∗ i ∗ ei i i=1 the Jacobi operator J is defined by (3.2) JV := ∆V − R(V )(V ∈ Γ(π−1TM)), ∆ is the rough Laplacian defined by p X (3.3) ∆V := − {∇ (∇ V ) − ∇ V }, ei ei ∇ei ei i=1 and h (3.4) R(V ) := R (V, π∗ei)π∗ei, p where {ei}i=1 is a locally defined orthonormal frame field on (P, g). The tangent space Pu (u ∈ P ) is canonically decomposed into the orthogonal ∗ direct sum of the vertical subspace Gu = {Au | A ∈ g} and the horizontal subspace Hu: Pu = Gu ⊕ Hu. Then, we have p X h τ2(π) = ∆τ(π) − R (τ(π), π∗ei)π∗ei i=1 m X h = ∆τ(π) − R (τ(π), π∗ei)π∗ei i=1 k X h ∗ ∗ − R (τ(π), π∗Am+i)π∗Am+i i=1 m X h = ∆τ(π) − R (τ(π), π∗ei)π∗ei, i=1 where p = dim P, m = dim M, k = dim G, respectively. Then, we obtain Z 0 = hJ(τ(π)), τ(π)i vg M Z Z m ∗ X h = h∇ ∇ τ(π), τ(π)i vg − hR (τ(π), π∗ei)π∗ei, τ(π)i vg M M i=1 Z Z m X h = h∇ τ(π), ∇ τ(π)i vg − hR (τ(π), π∗ei)π∗ei, τ(π)i vg. M M i=1 Therefore, we obtain Z Z m X h h∇ τ(π), ∇ τ(π)i vg = hR (τ(π), π∗ei)π∗ei, τ(π)i vg M M i=1 BIHARMONIC MAPS ON PRINCIPAL BUNDLES & WARPED PRODUCTS 563

Z m X h 0 0 = hR (τ(π), ei)ei, τ(π)i vg M i=1 Z h (3.5) = Ric (τ(π)) vg, M 0 m where {ei}i=1 is a locally deﬁned orthonormal frame ﬁeld on (M, h) satisfying 0 π∗ei = ei, and Ric(X) is the Ricci curvature of (M, h) along X ∈ TxM. The left hand side of (3.5) is non-negative, and then, the both hand sides of (3.5) must vanish if the Ricci curvature of (M, h) is non-positive. Therefore, we obtain ( ∇X τ(π) = 0 (∀X ∈ X(P )), i.e., τ(π) is parallel, and (3.6) Rich(τ(π)) = 0.

Let us deﬁne a 1-form α ∈ A1(P ) on P by α(X) = hdπ(X), τ(π)i,X ∈ X(P ). Then, we have p X (3.7) −δα = (∇ei α)(ei) = hτ(π), τ(π)i + hdπ, ∇τ(π)i. i=1 Integrate the above (3.7) over P since P is compact without boundary. By (3.6), ∇X τ(π) = 0,X ∈ X(P ), we have Z Z (3.8) 0 = − δα vg = hτ(π), τ(π)i vg P P which implies that τ(π) = 0, i.e., π :(P, g) → (M, h) is harmonic.

4. Proof of Theorem 4.1 In this section, we will show: Theorem 4.1. Let π :(P, g) → (M, h) be a principal G-bundle over a Rie- mannian manifold with non-positive Ricci curvature. Assume that (P, g) is a non-compact complete Riemannian manifold, and the projection π has both finite energy E(π) < ∞ and finite bienergy E2(π) < ∞. If π is biharmonic, then it is harmonic. Here, we first recall the following examples:

Example 1 (cf. [29], p. 62). The inversion in the unit sphere φ : Rn\{o} 3 x n x 7→ |x|2 ∈ R is a biharmonic morphism if n = 4. It is not harmonic since 4 x τ(φ) = − |x|4 . Here, a C∞ map φ :(M, g) → (N, h) is called to be a biharmonic morphism if, for every biharmonic function f : U ⊂ N → R with φ−1(U) 6= ∅, the composition f ◦ φ : φ−1(U) ⊂ M → R is biharmonic. 564 H. URAKAWA

Example 2 (cf. [29], p. 70). Let (M 2, h) be a Riemannian surface, and let β : M 2 × R → R∗ and λ : R → R∗ be two positive C∞ functions. Consider the projection π :(M 2 × R∗, g = λ−2 h + β2 dt2) 3 (p, t) 7→ p ∈ (M 2, h). c1x R f(x) dx −c1 (1+e ) ∗ Here, we take β = c2 e , f(x) = 1−ec1x with c1, c2 ∈ R , and (M 2, h) = (R2, dx2 + dy2). Then, 2 ∗ 2 2 2 2 2 2 2 π :(R × R , dx + dy + β (x) dt ) 3 (x, y, t) 7→ (x, y) ∈ (R , dx + dy ) gives a family of proper biharmonic (i.e., biharmonic but not harmonic) Rie- mannian submersions. For a non-compact and complete Riemannian manifold (N, h) with non- positive Ricci curvature, we will give a proof of Theorem 4.1. (The first step) We first take a cut off function η on (P, g) for a fixed point p0 ∈ P as follows: 0 ≤ η ≤ 1 (on P ),  η = 1 (on B (p )),  r 0 (4.1) η = 0 (outside B2r(p0)),   2 |∇η| ≤ (on P ), r where Br(p0) is the ball in (P, g) of radius r around p0. Now assume that the projection π :(P, g) → (N, h) is biharmonic. Namely, we have, by definition,

0 = J2(π) = Jπ(τ(π)) p X h (4.2) = ∆ τ(π) − R (τ(π), π∗ei)π∗ei, i=1 p where {ei}i=1 is a local orthonormal frame field on (P, g) and ∆ is the rough Laplacian which is defined by p ∗ X (4.3) ∆V := ∇ ∇V = − ∇ (∇ V ) − ∇ V ei ei ∇ei ei i=1 for V ∈ Γ(π−1TM). (The second step) By (4.2), we have Z Z * p + ∗ 2 2 X h h∇ ∇ τ(π), η τ(π)i vg = η R (τ(π), π∗ei)π∗ei, τ(π) vg P P i=1 Z p 2 X h = η R (τ(π), π∗ei)π∗ei, τ(π) vg P i=1 Z m 2 X h 0 0 = η R (τ(π), e i)e i, τ(π) vg P i=1 BIHARMONIC MAPS ON PRINCIPAL BUNDLES & WARPED PRODUCTS 565 Z 2 h (4.4) = η Ric (τ(π)) vg, P 0 m h where {e i}i=1 is a local orthonormal frame field on (M, h), and Ric (u) u ∈ TyM,(y ∈ M) is the Ricci curvature of (M, h) which is non-positive by our assumption. (The third step) Therefore, we obtain Z ∗ 2 0 ≥ h∇ ∇ τ(π), η τ(π)i vg P Z 2 = h∇ τ(π), ∇(η τ(π))i vg P Z p X 2 = h∇ei τ(π), ∇ei (η τ(π))i vg P i=1 Z p X 2 2 = η h∇ei τ(π), ∇ei τ(π)i + ei(η ) h∇ei τ(π), τ(π)i vg P i=1 Z p Z p 2 X 2 X (4.5) = η |∇ei τ(π)| vg + 2 hη ∇ei τ(π), ei(η) τ(π)i vg. P i=1 P i=1 Therefore, we obtain by (4.5). (The fourth step) Then, we have Z p Z p 2 X 2 X η ∇ei τ(π) vg ≤ −2 hη ∇ei τ(π), ei(η) τ(π)i vg P i=1 P i=1 p Z X (4.6) = −2 hVi,Wii vg, P i=1 where Vi := η ∇ei τ(π), and Wi := ei(η) τ(π)(i = 1, . . . , p). Then, the right hand side of (4.6) is estimated by the Cauchy-Schwarz inequality, 1 (4.7) ± 2 hV ,W i ≤ |V |2 + |W |2 i i i i since √ 1 1 0 ≤ | V ± √ W |2 = |V |2 ± 2 hV ,W i + |W |2, i i i i i i so that 1 ∓ 2 hV ,W i ≤ |V |2 + |W |2. i i i i Therefore, the right hand side of (4.6) is estimated as follows: p Z X RHS of (4.6) := − hVi,Wii vg P i=1 p p Z X 1 Z X (4.8) ≤ |V |2 v + |W |2 v . i g i g P i=1 P i=1 566 H. URAKAWA

1 (The ﬁfth step) By putting = 2 , we have p p p Z X 1Z X Z X (4.9) η2 |∇ τ(π)|2v ≤ η2|∇ τ(π)|2v +2 e (η)2|τ(π)|2v . ei g 2 ei g i g P i=1 P i=1 P i=1 Therefore, we obtain p 1 Z X Z (4.10) η2 |∇ τ(π)|2 v ≤ 2 |∇η|2 |τ(π)|2 v . 2 ei g g P i=1 P Substituting (4.1) into (4.10), we obtain p Z X Z 16 Z (4.11) η2 |∇ τ(π)|2 v ≤ 4 |∇η|2 |τ(π)|2 v ≤ |τ(π)|2 v . ei g g r2 g P i=1 P P

(The sixth step) Tending r → ∞ by the completeness of (P, g) and E2(π) = 1 R 2 2 P |τ(π)| vg < ∞, we obtain that Z p X 2 (4.12) |∇ei τ(π)| vg = 0, P i=1 which implies that

(4.13) ∇X τ(π) = 0 (∀ X ∈ X(P )). (The seventh step) Therefore, we obtain (4.14) |τ(π)| is constant, say c because 2 X |τ(π)| = 2 h∇X τ(π), τ(π)i = 0 (∀ X ∈ X(M)) by (4.13). (The eighth step) In the case that Vol(P, g) = ∞ and E2(π) < ∞, c must be zero. Because, if c 6= 0, Z 1 2 c E2(π) = |τ(π)| vg = Vol(P, g) = ∞ 2 P 2 which is a contradiction. Thus, if Vol(P, g) = ∞, then c = 0, i.e., π :(P, g) → (M, h) is harmonic. (The ninth step) In the case E(π) < ∞ and E2(π) < ∞, let us deﬁne a 1-form α ∈ A1(P ) on P by (4.15) α(X) := hdπ(X), τ(π)i, (X ∈ X(P )). Then, we obtain Z Z p 1/2 Z X 2 |α| vg = |α(ei)| ≤ |dπ| |τ(π)| vg P P i=1 P Z 1/2 Z 1/2 2 2 ≤ |dπ| vg |τ(π)| vg P P BIHARMONIC MAPS ON PRINCIPAL BUNDLES & WARPED PRODUCTS 567 p (4.16) = 2 E(π) E2(π) < ∞.

Pp ∞ For the function δα := − i=1(∇ei α)(ei) ∈ C (P ), we have p p X X −δα = (∇ei α)(ei) = ei(α(ei)) − α(∇ei ei) i=1 i=1 p X = ei hdπ(ei), τ(π)i − hdπ(∇ei ei), τ(π)i i=1 p X = ∇ei dπ(ei), τ(π) + dπ(ei), ∇ei τ(π) − dπ(∇ei ei, τ(π) i=1 p p X X = ∇ei dπ(ei) − dπ(∇ei ei) , τ(π) + dπ(ei), ∇ei τ(π) i=1 i=1 = hτ(π), τ(π)i + hdπ, ∇τ(π)i (4.17) = |τ(π)|2 since ∇τ(π) = 0. By (4.17), we obtain Z Z 2 (4.18) |δα| vg = |τ(π)| vg = 2 E2(π) < ∞. P P By (4.16), (4.18) and the completeness of (P, g), we can apply Gaﬀney’s theorem which implies that Z Z 2 (4.19) 0 = (−δ α) vg = |τ(π)| vg. P P Thus, we obtain (4.20) τ(π) = 0, that is, π :(P, g) → (M, h) is harmonic. We obtain Theorem 4.1.

5. The tension fields of the warped products In this section, we calculate the tension field τ(π). Let us recall the definition of the tension field: Definition 5.1. m+` X g τ(π) = ∇ei π∗ei − π∗ (∇ ei ei) i=1 m+` X h g (5.1) = ∇ π∗ei π∗ei − π∗ (∇ ei ei) . i=1 568 H. URAKAWA

g h 0 Since ∇ e are the horizontal lifts of ∇ 0 e for i = 1, . . . , m, and (2.14), ei i ei i we have m X h g τ(π) = ∇ π∗ei π∗ei − π∗ (∇ ei ei) i=1 m+` X h g + ∇ π∗ei π∗ei − π∗ (∇ ei ei) i=m+1 m m+` X h 0 h 0 X 1 = ∇ e0 e − ∇ e0 e + 0 − − ∇ (f ◦ π) i i i i f i=1 i=m+1 ` (5.2) = ∇ (f ◦ π). f Indeed, we obtain the second equality of (5.2) as follows: The ﬁrst sum van- 0 g h 0 ishes since π∗ei = e and π∗ ∇ ei ei = ∇ 0 e ,(i = 1, . . . , m). The second sum i ei i ` g 1 coincides with f ∇ (f ◦ π) since π∗ei = 0 and also π∗ ∇ ei ei = − f ∇ (f ◦ π) (i = m + 1, . . . , m + `). Therefore, we obtain: Theorem 5.1. Let π :(P, g) → (M, h) be the warped product. Then, we have ` (5.3) τ(π) = ∇ (f ◦ π). f Then, π is harmonic if and only if f is constant.

6. The bitension fields of the warped products Let us recall the definition of the bitension field for a C∞ mapping ϕ : (P, g) → (M, h) which is given by

h (6.1) τ2(ϕ) := ∆ τ(ϕ) − R (τ(ϕ)). Here, recall, for V ∈ Γ(ϕ−1TM), p X (6.2) ∆V := − ∇ (∇ V ) − ∇ g V , ei ei ∇ ei ei i=1 p h X h (6.3) R V := R (V, ϕ∗ei)ϕ∗ei, i=1 p where {ei}i=1 is a locally deﬁned orthonormal frame ﬁeld on (P, g), p = dim P , ∇ is the induced connection on the induced bundle ϕ−1TM, and the curvature h h h h h tensor of (N, h) is given by R (U, V )W := ∇ U (∇ V W ) − ∇ V (∇ U W ) − h ∇ [U,V ]W for U, V, W ∈ X(M) (cf. [5, 21, 22]).

Deﬁnition 6.1. π :(P, g) → (M, h) is biharmonic if τ2(π) = 0. BIHARMONIC MAPS ON PRINCIPAL BUNDLES & WARPED PRODUCTS 569

Let us π :(P, g) → (M, h) be the warped product whose Riemannian metric g is given by (2.8). For V = τ(π), then, p h X h R V = R (τ(π), π∗ei)π∗ei i=1 m X h 0 0 = R (τ(π), ei)ei i=1 (6.4) = ρh(τ(π)), where m = dim M and ρh is Ricci transform of (M, h) given by ρh(u) := Pm h 0 0 0 m i=1 R (u, ei)ei, u ∈ TxM, and {ei}i=1 is a locally deﬁned orthonormal ﬁeld on (M, h). In the following, we calculate the rough Laplacian ∆ for V = τ(π).

(The ﬁrst step) We calculate ∇ei τ(π) and ∇ei (∇ei τ(π)) as follows: ( h ∇ 0 τ(π)(i = 1, . . . , m = dim M), h ei (6.5) ∇e τ(π) = ∇ τ(π) = i π∗ei 0 (i = m + 1, . . . , m + `), where p := dim P = m + `, m = dim M, and ` = dim F . Furthermore, ( h h ∇ 0 (∇ 0 τ(π)) (i = 1, . . . , m), ei ei (6.6) ∇e (∇e τ(π)) = i i 0 (i = m + 1, . . . , m + ` = p).

(The second step) We calculate ∇ g τ(π) by the similar way as the ﬁrst ∇ei ei step: For i = 1, . . . , m, h h (6.7) ∇ g τ(π) = ∇ g τ(π) = ∇ h 0 τ(π), ∇e ei π∗(∇e ei) ∇ e i i e0 i i and for i = m + 1, . . . , m + `, by (5.1), h h g g 1 (6.8) ∇∇e ei τ(π) = ∇ τ(π) = ∇ τ(π). i π∗(∇ei ei) − f ∇(f ◦ π) (The third step) Therefore, we calculate (6.2) for V = τ(π) as follows. p X n o ∆τ(π) := − ∇e (∇e τ(π)) − ∇ g τ(π) i i ∇ei ei i=1 m X n o = − ∇e (∇e τ(π)) − ∇ g τ(π) i i ∇ei ei i=1 m+` X n o − ∇e (∇e τ(π)) − ∇ g τ(π) i i ∇ei ei i=m+1 m X n h h h o = − ∇ 0 (∇ 0 τ(π)) − ∇ h 0 τ(π) ei ei ∇ e0 e i i i=1 570 H. URAKAWA

m+` X n h o − 0 − ∇ 1 τ(π) − f ∇(f ◦ π) i=m+1 m X h h h h (6.9) = − ∇ e0 ∇ e0 τ(π) − ∇ ∇h e0 τ(π) − ` ∇ 1 ∇(f ◦ π)τ(π). i i e0 i f i=1 i (The fourth step) Therefore, by (6.1), (6.4) and (6.9), we obtain m X n o h h τ (π) = − ∇ 0 ∇ 0 τ(π) − ∇ h 0 τ(π) − ` ∇ 1 τ(π) − ρ (τ(π)) 2 ei ei ∇ e0 e ∇f i i f i=1 1 2 h ∇f (6.10) = ` Jid ∇f − ` ∇ 1 ∇f f f f 1 `2 `2 (6.11) = ` J ∇f + h(∇f, ∇f) ∇f − ∇h ∇f. id f f 3 f 2 ∇f Therefore, we can summarize the above by recalling the following deﬁnitions:

h h (6.12) Jid := ∆ − ρ , is the Jacobi operator of the identity of (M, h), id : (M, h) → (M, h) acting on h the space X(M) of C∞ vector fields on M, and the operator ∆ is defined by m h X h h h (6.13) ∆ (X) := − ∇ 0 ∇ 0 − ∇ h 0 X (X ∈ X(M)), ei ei ∇ e0 e i i i=1 and ρh is the Ricci operator of (M, h) given by h h 0 0 (6.14) ρ (X) = R (X, ei)ei (X ∈ X(M)). Therefore, due to (6.1) and (6.2), we have: Theorem 6.1. For the warped product π :(P, g) → (M, h), the bitension field τ2(π) is given by h (6.15) τ2(π) = ∆(τ(π)) − ρ (τ(π)) − ` ∇ ∇ f τ(π), f where ∆ is the rough Laplacian and ∇ is the induced connection from the Levi- Civita connection ∇h of (M, h). Therefore, the warped product π :(P, g) → (M, h) is biharmonic, i.e., τ2(π) = 0, if and only if the following hold: ∇f h ∇f ` ` h (6.16) Jid = ` ∇ 1 ∇f = − h(∇f, ∇f) ∇ f + ∇ ∇ f ∇f. f f f f 3 f 2 ∞ Corollary 6.2. For a positive C function f on M, let π :(P, g) = (M ×f F, g) → (M, h) be the warped product with g = π∗h + f 2 k over a Riemannian manifold (M, h) whose Ricci curvature is non-positive. If π is biharmonic, then Z Z ∇f ∇f h ∇f ∇f (6.17) (∇f) h( , ) vh = 2 h ∇ ∇f , vh ≥ 0. M f f M f f f BIHARMONIC MAPS ON PRINCIPAL BUNDLES & WARPED PRODUCTS 571

Proof of Corollary 6.2. If π :(P, g) → (M, h) is biharmonic, by (6.15), it holds that ∇f 2 h ∇f (6.18) 0 = τ2(π) = Jid ` − ` ∇ ∇f , f f f which implies that Z Z ∇f ∇f h ∇f ∇f (6.19) 0 ≤ h Jid , vh = ` h ∇ ∇f , vh. M f f M f f f h h Because all the eigenvalues of Jid are non-negative since Jid = ∆ − ρ and the h Ricci transform ρ are non-positive (cf. [4], [6, p. 161]). 7. The solutions of the ordinary diﬀerential equation 2 Assume that (M, h) = (R, dt ), a line, and (P, g) = F ×f R, the warped product of a Riemannian manifold (F, k) and the line (R, dt2), that is, (7.1) g = π∗(dt2) + f 2 k for a C∞ function f ∈ C∞(R). In this case, it holds that  ∇f f 0 00 ∂  J (π) = J ` = −` ,  id id  f f ∂t  0 0 0 0  2 h ∇f 2 f h f ∂ 2 f ∂ f ∂ (7.2) ` ∇ ∇f = ` ∇ ∂ = `  f f f ∂t f ∂t f ∂t f ∂t   f 0 f 00 f 03 ∂  = `2 − .  f 2 f 3 ∂t 2 Therefore, π :(F ×f R, g) → (R, dt ) is biharmonic, i.e., ∇f 2 h ∇f (7.3) τ2(π) = Jid ` − ` ∇ ∇f = 0 f f f if and only if f 0 00 f 0 f 00 f 03 0 = −` − `2 − f f 2 f 3 f 00f − f 02 0 f 0 f 00 f 03 = −` − `2 − f 2 f 2 f 3 f 000 f 2 − 3 f 00 f 0 f + 2 f 03 f 0 f 00 f 03 = −` − `2 − f 3 f 2 f 3 ` (7.4) = − f 000 f 2 + (` − 3) f 00 f 0 f + (−` + 2) f 03 f 3 if and only if (7.5) f 000 f 2 + (` − 3) f 00 f 0 f + (−` + 2) f 03 = 0. Therefore, we have: 572 H. URAKAWA

Theorem 7.1. Let (F, k) be a Riemannian manifold. 2 (1) the warped product π :(F ×f R, g) → (R, dt ) is biharmonic if and only if (7.5) holds. (2) All the positive C∞ solution f of (7.5) on R are given by Z t ` (7.6) f(t) = c exp a tanh a r + b dr , t0 2 where a 6= 0, b, c > 0 are arbitrary constants. (3) In the case (M, h) = (R, dt2), let f(t) be a C∞ function deﬁned by (7.6) with a 6= 0, b any real number and c > 0. Then, the warped product 2 π :(R ×f F, g) → (R, dt ) with the Riemannian metric (7.7) g = π∗dt2 + f 2 k is biharmonic but not harmonic.

0 f 0 In order to solve (7.5), we put u := (log f) = f . Then (7.5) turns into the ordinary differential equation on u: ` (7.8) u00 + (u2)0 = 0. 2 A general solution u of (5.6) is given by ` (7.9) u(t) = a tanh a t + b , 2 where a and b are arbitrary constants. Thus, every positive solution f(t) is given by Z t ` (7.10) f(t) = c exp a tanh a r + b dr , t0 2 where a, b, c > 0 are arbitrary constants. Therefore, we obtain Theorem 7.1 together with Theorem 6.1, References [1] K. Akutagawa and S. Maeta, Biharmonic properly immersed submanifolds in Euclidean spaces, Geom. Dedicata 164 (2013), 351–355. [2] A. Balmus, S. Montaldo, and C. Oniciuc, Classification results for biharmonic submanifolds in spheres, Israel J. Math. 168 (2008), 201–220. [3] , Biharmonic hypersurfaces in 4-dimensional space forms, Math. Nachr. 283 (2010), no. 12, 1696–1705. [4] R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49. [5] C. P. Boyer and K. Galicki, Sasakian Geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008. [6] R. Caddeo, S. Montaldo, and P. Piu, On biharmonic maps, in Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), 286–290, Contemp. Math., 288, Amer. Math. Soc., Providence, RI, 2000. [7] I. Castro, H. Li, and F. Urbano, Hamiltonian-minimal Lagrangian submanifolds in complex space forms, Pacific J. Math. 227 (2006), no. 1, 43–63. BIHARMONIC MAPS ON PRINCIPAL BUNDLES & WARPED PRODUCTS 573

[8] B. Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17 (1991), no. 2, 169–188. [9] F. Defever, Hypersurfaces of E4 with harmonic mean curvature vector, Math. Nachr. 196 (1998), 61–69. [10] J. Eells and L. Lemaire, Selected topics in harmonic maps, CBMS Regional Conference Series in Mathematics, 50, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1983. [11] N. Ejiri, A construction of nonflat, compact irreducible Riemannian manifolds which are isospectral but not isometric, Math. Z. 168 (1979), no. 3, 207–212. [12] D. Fetcu and C. Oniciuc, Biharmonic integral C-parallel submanifolds in 7-dimensional Sasakian space forms, Tohoku Math. J. (2) 64 (2012), no. 2, 195–222. 4 [13] Th. Hasanis and Th. Vlachos, Hypersurfaces in E with harmonic mean curvature vector field, Math. Nachr. 172 (1995), 145–169. [14] T. Ichiyama, J. Inoguchi, and H. Urakawa, Bi-harmonic maps and bi-Yang-Mills fields, Note Mat. 28 (2009), suppl. 1, 233–275. [15] , Classifications and isolation phenomena of bi-harmonic maps and bi-Yang- Mills fields, Note Mat. 30 (2010), no. 2, 15–48. [16] J.-I. Inoguchi, Submanifolds with harmonic mean curvature vector field in contact 3- manifolds, Colloq. Math. 100 (2004), no. 2, 163–179. m [17] H. Iriyeh, Hamiltonian minimal Lagrangian cones in C , Tokyo J. Math. 28 (2005), no. 1, 91–107. [18] S. Ishihara and S. Ishikawa, Notes on relatively harmonic immersions, Hokkaido Math. J. 4 (1975), no. 2, 234–246. [19] G. Y. Jiang, 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A 7 (1986), no. 4, 389–402. [20] T. Kajigaya, Second variation formula and the stability of Legendrian minimal submanifolds in Sasakian manifolds, Tohoku Math. J. (2) 65 (2013), no. 4, 523–543. [21] S. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, New York, 1972. [22] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York, 1963. [23] N. Koiso and H. Urakawa, Biharmonic submanifolds in a Riemannian manifold, accepted in Osaka J. Math. [24] Y. Luo, Weakly convex biharmonic hypersurfaces in nonpositive curvature space forms are minimal, Results Math. 65 (2014), no. 1-2, 49–56. [25] , On biharmonic submanifolds in non-positively curved manifolds, J. Geom. Phys. 88 (2015), 76–87. [26] , Remarks on the nonexistence of biharmonic maps, Arch. Math. (Basel) 107 (2016), no. 2, 191–200. [27] E. Loubeau and C. Oniciuc, The index of biharmonic maps in spheres, Compos. Math. 141 (2005), no. 3, 729–745. [28] , On the biharmonic and harmonic indices of the Hopf map, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5239–5256. [29] E. Loubeau and Y.-L. Ou, Biharmonic maps and morphisms from conformal mappings, Tohoku Math. J. (2) 62 (2010), no. 1, 55–73. [30] S. Maeta and H. Urakawa, Biharmonic Lagrangian submanifolds in Kählermanifolds, Glasg. Math. J. 55 (2013), no. 2, 465–480. [31] S. Montaldo and C. Oniciuc, A short survey on biharmonic maps between Riemannian manifolds, Rev. Un. Mat. Argentina 47 (2006), no. 2, 1–22. [32] Y. Nagatomo, Harmonic maps into Grassmannians and a generalization of do Carmo- Wallach theorem, in Riemann surfaces, harmonic maps and visualization, 41–52, OCAMI Stud., 3, Osaka Munic. Univ. Press, Osaka, 2008. 574 H. URAKAWA

[33] H. Naito and H. Urakawa, Conformal change of Riemannian metrics and biharmonic maps, Indiana Univ. Math. J. 63 (2014), no. 6, 1631–1657. [34] N. Nakauchi and H. Urakawa, Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature, Ann. Global Anal. Geom. 40 (2011), no. 2, 125–131. [35] , Biharmonic submanifolds in a Riemannian manifold with non-positive curvature, Results Math. 63 (2013), no. 1-2, 467–474. [36] N. Nakauchi, H. Urakawa, and S. Gudmundsson, Biharmonic maps into a Riemannian manifold of non-positive curvature, Geom. Dedicata 169 (2014), 263–272. [37] S. Ohno, T. Sakai, and H. Urakawa, Rigidity of transversally biharmonic maps between foliated Riemannian manifolds, Hokkaido Math. J., 2017. [38] C. Oniciuc, Biharmonic maps between Riemannian manifolds, An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si.Mat. (N.S.) 48 (2002), no. 2, 237–248. [39] Y.-L. Ou and L. Tang, The generalized Chen’s conjecture on biharmonic submanifolds is false, arXiv: 1006.1838v1. [40] , On the generalized Chen’s conjecture on biharmonic submanifolds, Michigan Math. J. 61 (2012), no. 3, 531–542. [41] T. Sasahara, Legendre surfaces in Sasakian space forms whose mean curvature vectors are eigenvectors, Publ. Math. Debrecen 67 (2005), no. 3-4, 285–303. [42] , Stability of biharmonic Legendrian submanifolds in Sasakian space forms, Canad. Math. Bull. 51 (2008), no. 3, 448–459. [43] , A class of biminimal Legendrian submanifolds in Sasakian space forms, Math. Nachr. 287 (2014), no. 1, 79–90. [44] R. T. Smith, The second variation formula for harmonic mappings, Proc. Amer. Math. Soc. 47 (1975), 229–236. [45] T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380–385. [46] K. Tsukada, Eigenvalues of the Laplacian of warped product, Tokyo J. Math. 3 (1980), no. 1, 131–136. [47] H. Urakawa, Calculus of Variations and Harmonic Maps, translated from the 1990 Japanese original by the author, Translations of Mathematical Monographs, 132, Amer- ican Mathematical Society, Providence, RI, 1993. [48] , CR rigidity of pseudo harmonic maps and pseudo biharmonic maps, Hokkaido Math. J. 46 (2017), no. 2, 141–187. [49] Z.-P. Wang and Y.-L. Ou, Biharmonic Riemannian submersions from 3-manifolds, Math. Z. 269 (2011), no. 3-4, 917–925.

Hajime Urakawa Global Learning Center Tohoku University Kawauchi 41, Sendai 980-8576, Japan Email address: urakawa@@math.is.tohoku.ac.jp