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

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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 1 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(') = 2 M jd'j vg for smooth maps ' 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('). 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(') = jτ(')j 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 subman- ifold 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 con- jecture, 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 com- pact 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 fi- nite energy E(π) < 1 and finite bienergy E2(π) < 1. 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 C1 function f on M, f 2 C1(M), the warping function on M, let us consider the warped product (P; g) where π : P = M × F 3 (x; y) 7! x 2 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 2 C1(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 2 C1(M). Then, the tension field τ(π) is given by grad f r 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)) − ` r r f τ(π); f where ∆ is the rough Laplacian and r is the induced connection from the Levi- Civita connection rh of (M; h). Therefore, π is biharmonic if and only if h (1.4) ∆(τ(π)) − ρ (τ(π)) − ` r r 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 2 TxM for an 0 m locally defined orthonormal field feigi=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 2 C1(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 C1 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 ' :(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(') = e(') vg; M 556 H. URAKAWA 1 2 where e(') := 2 jd'j is called the energy density of '. That is, for any variation f'tg of ' with '0 = ', Z d (2.1) E('t) = − h(τ(');V )vg = 0; dt t=0 M where V 2 Γ('−1TN) is a variation vector field along ' which is given by d V (x) = dt t=0't(x) 2 T'(x)N,(x 2 M), and the tension field is given by Pm −1 m τ(') = i=1 B(')(ei; ei) 2 Γ(' TN), where feigi=1 is a locally defined orthonormal frame field on (M; g), and B(') is the second fundamental form of ' defined by B(')(X; Y ) = (re d')(X; Y ) = (re X d')(Y ) = rX (d'(Y )) − d'(rX Y )(2.2) for all vector fields X; Y 2 X(M). Here, r, and rh, are Levi-Civita connections on TM, TN of (M; g), (N; h), respectively, and r, and re are the induced ones on '−1TN, and T ∗M ⊗ '−1TN, respectively. By (2.1), ' is harmonic if and only if τ(') = 0. The second variation formula is given as follows. Assume that ' is harmonic. Then, 2 Z d (2.3) 2 E('t) = h(J(V );V )vg; dt t=0 M where J is an elliptic differential operator, called the Jacobi operator acting on Γ('−1TN) given by (2.4) J(V ) = ∆V − R(V ); ∗ where ∆V = r rV = − Pm fr r V − r V g is the rough Laplacian i=1 ei ei rei ei and R is a linear operator on Γ('−1TN) given by m X N R(V ) = R (V; d'(ei))d'(ei); i=1 N h h h and R is the curvature tensor of (N; h) given by R (U; V ) = r U r V − h h h r V r U − r [U;V ] for U; V 2 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(') = jτ(')j vg; 2 M where jV j2 = h(V; V ), V 2 Γ('−1TN).
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