165 TANGENT BUNDLE OF ORDER TWO AND BIHARMONICITY 1 ...

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Acta Math. Univ. Comenianae Vol. LXXXIII, 2 (2014), pp. 165–179

TANGENT BUNDLE OF ORDER TWO AND BIHARMONICITY H. ELHENDI, M. TERBECHE and M. DJAA Abstract. The problem studied in this paper is related to the biharmonicity of a section from a Riemannian manifold (M, g) to its tangent bundle T 2 M of order two equipped with the diagonal metric g D . We show that a section on a compact manifold is biharmonic if and only if it is harmonic. We also investigate the curvature of (T 2 M, g D ) and the biharmonicity of section of M as a map from (M, g) to (T 2 M, g D ).

1. Introduction Harmonic (resp., biharmonic) maps are critical points of energy (resp., bienergy) functional defined on the space of smooth maps between Riemannian manifolds introduced by Eells and Sampson [4] (resp., Jiang [6]). In this paper, we present some properties for biharmonic section between a Riemannian manifold and its second tangent bundle which generalize the results of Ishihara [5], Konderak [7], Oproiu [9] and Djaa-Ouakkas [3]. Consider a smooth map φ : (M n , g) → (N n , h) between two Riemannian manifolds, then the energy functional is defined by Z 1 E(φ) = (1) |dφ|2 dvg 2 M (or over any compact subset K ⊂ M ). A map is called harmonic if it is a critical point of the energy functional E (or E(K) for all compact subsets K ⊂ M ). For any smooth variation {φ}t∈I of φ t with φ0 = φ and V = dφ dt |t=0 , we have Z d 1 (2) E(φt )|t=0 = − h(τ (φ), V )dvg , dt 2 M where (3)

τ (φ) = trg ∇dφ

is the tension field of φ. Then we have the following theorem. Received February 24, 2013; revised January 28, 2014. 2010 Mathematics Subject Classification. Primary 53A45, 53C20, 58E20. Key words and phrases. Diagonal metric; λ-lift; biharmonic section. The authors was supported by LGACA and GMFAMI Laboratories. The authors would like to thank the referee for his useful remarks and suggestions.

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H. ELHENDI, M. TERBECHE and M. DJAA

Theorem 1.1. A smooth map φ : (M m , g) → (N n , h) is harmonic if and only if (4)

τ (φ) = 0. i

α

If (x )1≤i≤m and (y )1≤α≤n denote local coordinates on M and N , respectively, then equation (4) takes the form β γ N α α ij α ∂φ ∂φ (5) τ (φ) = ∆φ + g Γβγ = 0, ∂xi ∂xj N p α ∂ m α where ∆φα = √1 ∂x |g|g ij ∂φ i( ∂xj ) is the Laplace operator on (M , g) and Γβγ |g|

are the Christoffel symbols on N . Definition 1.2. A map φ : (M, g) → (N, h) between Riemannian manifolds is called biharmonic if it is a critical point of bienergie functional Z 1 E2 (φ) = (6) |τ (φ)|2 dv g . 2 M The Euler-Lagrange equation attached to bienergy is given by vanishing of the bitension field (7)

τ2 (φ) = −Jφ (τ (φ)) = −(∆φ τ (φ) + trg RN (τ (φ), dφ)dφ),

where Jφ is the Jacobi operator defined by (8)

Jφ : Γ(φ−1 (T N )) → Γ(φ−1 (T N )) V 7→ ∆φ V + trg RN (V, dφ)dφ.

Theorem 1.3. A smooth map φ : (M m , g) → (N n , h) is biharmonic if and only if (9)

τ2 (φ) = 0.

From Theorem 1.1 and formula (7), we have the following corollary. Corollary 1.4. If φ : (M m , g) → (N n , h) is harmonic, then φ is biharmonic. (For more details see [6]). 2. Preliminary Notes 2.1. Horizontal and vertical lifts on TM Let (M, g) be an n-dimensional Riemannian manifold and (T M, π, M ) be its tangent bundle. A local chart (U, xi )i=1...n on M induces a local chart (π −1 (U ), xi , y j )i,j=1,...,n on T M . Denote the Christoffel symbols of g by Γkij and the Levi-Civita connection of g by ∇. We have two complementary distributions on T M , the vertical distribution V and the horizontal distribution H defined by V(x,u) = Ker(dπ(x,u) ) ∂ = ai i (x,u) ; ai ∈ R , ∂y ∂ ∂ H(x,u) = ai i (x,u) − ai uj Γkij k (x,u) ; ai ∈ R , ∂x ∂y

TANGENT BUNDLE OF ORDER TWO AND BIHARMONICITY

167

where (x, u) ∈ T M , such that T(x,u) T M = H(x,u) ⊕ V(x,u) . ∂ Let X = X i ∂x i be a local vector field on M . The vertical and the horizontal lifts of X are defined by ∂ ∂y i ∂ i j k ∂ i δ =X − y Γij k . =X δxi ∂xi ∂y

(10)

XV = Xi

(11)

XH

For consequences, we have: ∂ H ∂ V δ ∂ 1. = and = . ∂xi δxi ∂xi ∂y i δ ∂ 2. , j is a local frame on T M i δx ∂y i,j=1,...,n ∂ ∂ j k ∂ − y Γ and 3. If u = ui i ∈ Tx M , then uH = ui ij ∂x ∂xi ∂y k

uV = ui

∂ . ∂y i

Definition 2.1. Let (M, g) be a Riemannian manifold and F : T M → T M be a smooth bundle endomorphism of T M . Then we define a vertical and horizontal vector fields V F , HF on T M by V F : TM → TTM (x, u) 7→ (F (u))V , HF : T M → T T M (x, u) 7→ (F (u))H . Locally we have (12) (13)

V F = y i Fij HF = y

i

Fij

V ∂ ∂ i = y F ∂y j ∂xi H ∂ ∂ ∂ i k j s i − y y Fi Γjk s = y F . ∂xj ∂y ∂xi

b be the Proposition 2.2 ([1]). Let (M, g) be a Riemannian manifold and ∇ s Levi-Civita connection of the tangent bundle (T M, g ) equipped with the Sasaki metric. If F is a tensor field of type (1, 1) on M , then b X V V F )(x,u) = (F (X))V , (∇ (x,u) b X V HF )(x,u) = (F (X))H + 1 (Rx (u, Xx )F (u))H , (∇ (x,u) 2 b X H V F )(x,u) = V (∇X F )(x, u) + 1 (Rx (u, Fx (u))Xx )H , (∇ 2 b X H HF )(x,u) = H(∇X F )(x, u) − 1 (Rx (Xx , Fx (u))u)V , (∇ 2 where (x, u) ∈ T M and X ∈ Γ(T M ).

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2.2. Second Tangent Bundle Let M be an n-dimensional smooth differentiable manifold and (Uα , ψα )α∈I a corresponding atlas. For each x ∈ M , we define an equivalence relation on Cx = {γ : (−ε, ε) → M ; γ is smooth and γ(0) = x, ε > 0} by γ ≈x h ⇔ γ 0 (0) = h0 (0)

and γ 00 (0) = h00 (0),

where γ 0 and γ 00 denote the first and the second derivation of γ, respectively, γ 0 : (−ε, ε) → T M ;

t 7→ [dγ(t)](1)

00

γ : (−ε, ε) → T (T M );

t 7→ [dγ 0 (t)](1).

Definition 2.3. We define the second tangent space of M at the point x to be the quotient Tx2 M = Cx / ≈x and S the second tangent bundle of M the union of all second tangent space, T 2 M = x∈M Tx2 M . We denote the equivalence class of γ by jx2 γ with respect to ≈x , and by j 2 γ an element of T 2 M . In the general case, the structure of higher tangent bundle T r M is considered in [8, Chapters 1–2] and [2]. Proposition 2.4 ([3]). Let M be an n-dimensional manifold, then T M is subbundle of T 2 M and the map i : T M → T 2M j 1 f = j 2 f˜

(14)

x

x

is an injective homomorphism of natural bundles (not of vector bundles), where Z t f˜i = (15) f i (s)ds − tf i (0) + f i (0) i = 1 . . . n. 0

Theorem 2.5. Let (M, g) be a Riemannian manifold and ∇ be the Levi-Civita connection. If T M ⊕ T M denotes the Whitney sum, then S : T 2M → T M ⊕ T M (16)

˙ j 2 γ(0) 7→ (γ(0), ˙ (∇γ(0) ˙ γ)(0))

is a diffeomorphism of natural bundles. In the induced coordinate we have (17)

(xi ; y i ; z i ) 7→ (xi , y i , z i + y j y k Γijk ).

Remark 2.6. The diffeomorphism S determines a vector bundle structure on T 2 M by α.Ψ1 + β · Ψ2 = S −1 (αS(Ψ1 ) + βS(Ψ2 )), where Ψ1 , Ψ2 ∈ T 2 M and α, β ∈ R, for which S is a linear isomorphism of vector bundles and i : T M → T 2 M is an injective linear homomorphism of vector bundles (for more details see [2]).


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Definition 2.7 ([3]). Let (M, g) be a Riemannian manifold and T 2 M be its tangent bundle of order two endowed with the vectorial structure induced by the diffeomorphism S. For any section σ ∈ Γ(T 2 M ), we define two vector fields on M by (18)

Xσ = P1 ◦ S ◦ σ,

(19)

Yσ = P2 ◦ S ◦ σ,

where P1 and P2 denote the first and the second projections from T M ⊕ T M on TM. From Remark 2.6 and Definition 2.7, we deduce the following. Proposition 2.8. For all sections σ, $ ∈ Γ(T 2 M ) and α ∈ R, we have Xασ+$ = αXσ + X$ , Yασ+$ = αYσ + Y$ , where ασ + $ = S −1 (αS(σ) + S($)). Definition 2.9 ([3]). Let (M, g) be a Riemannian manifold and T 2 M be its tangent bundle of order two endowed with the vectorial structure induced by the b on Γ(T 2 M ) by diffeomorphism S. We define a connection ∇ b : Γ(T M ) × Γ(T 2 M ) → Γ(T 2 M ) ∇ (20) b Z σ = S −1 (∇Z Xσ , ∇Z Yσ ) (z, σ) 7→ ∇ where ∇ is the Levi-Civita connection on M . Proposition 2.10. If (U, xi ) is a chart on M and (σ i , σ ¯ i ) are the components of section σ ∈ Γ(T 2 M ), then ∂ (21) Xσ = σ i i ∂x ∂ (22) Yσ = (¯ σ k + σ i σ j Γkij ) k . ∂x Proposition 2.11. Let (M, g) be a Riemannian manifold and T 2 M be its tangent bundle of order two, then (23)

J : Γ(T M ) → Γ(T 2 M ) Z 7→ S −1 (Z, 0)

is an injective homomorphism of vector bundles. Locally if (U ; xi ) is a chart on M and (U ; xi ; y i ) and (U ; xi ; y i ; z i ) are the induced charts on T M and T 2 M . respectively, then we have (24)

J : (xi , y i ) 7→ (xi , y i , −y j y k Γijk ).

Definition 2.12. Let (M, g) be a Riemannian manifold and X ∈ Γ(T M ) be a vector field on M . For λ = 0, 1, 2, the λ-lift of X to T 2 M is defined by (25)

X 0 = S∗−1 (X H , X H )

(26)

X 1 = S∗−1 (X V , 0)

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X 2 = S∗−1 (0, X V ).

(27)

Theorem 2.13 ([2]). Let (M, g) be a Riemannian manifold and R its tensor curvature, then for all vector fields X, Y ∈ Γ(T M ) and p ∈ T 2 M , we have: 1. [X 0 , Y 0 ]p = [X, Y ]0p − (R(X, Y )u)1 − (R(X, Y )w)2 , 2. [X 0 , Y i ] = (∇X Y )i , 3. [X i , Y j ] = 0, where (u, w) = S(p) and i, j = 1, 2. Definition 2.14. Let (M, g) be a Riemannian manifold. For any section σ ∈ Γ(T 2 M ), we define the vertical lift of σ to T 2 M by (28)

σ V = S∗−1 (XσV , YσV ) ∈ Γ(T (T 2 M )).

Remark 2.15. From Definition 2.7 and the formulae (14), (23) and (28), we obtain σ V = Xσ1 + Yσ2 , b Z σ)V = (∇Z Xσ )1 + (∇Z Yσ )2 , (∇ Z 1 = J(Z)V , Z 2 = i(Z)V for all σ ∈ Γ(T 2 M ) and Z ∈ Γ(T M ). 2.3. Diagonal metric Theorem 2.16 ([3]). Let (M, g) be a Riemannian manifold and T M its tangent bundle equipped with the Sasakian metric g s , then g D = S∗−1 (˜ g , g˜) is the only metric that satisfies the following formulae (29)

g D (X i , Y j ) = δij · g(X, Y ) ◦ π2

for all vector fields X, Y ∈ Γ(T M ) and i, j = 0, . . . , 2, where g˜ is the metric defined by 1 g˜(X H , Y H ) = g s (X H , Y H ), 2 g˜(X H , Y V ) = g s (X H , Y V ), g˜(X V , Y V ) = g s (X V , Y V ). g D is called the diagonal lift of g to T 2 M . e be the LeviProposition 2.17. Let (M, g) be a Riemannian manifold and ∇ Civita connection of the tangent bundle of order two equipped with the diagonal metric g D . Then: e X 0 Y 0 )p = (∇X Y )0 − 1 (R(X, Y )u)1 − 1 (R(X, Y )w)2 , 1. (∇ 2 2 e X 0 Y 1 )p = (∇X Y )1 + 1 (R(u, Y )X)0 , 2. (∇ 2

e X 0 Y 2 )p = (∇X Y )2 + 1 (R(w, Y )X)0 , 3. (∇ 2


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e X 1 Y 0 )p = 1 (Rx (u, X)Y )0 , 4. (∇ 2 e X 2 Y 0 )p = 1 (Rx (w, X)Y )0 , 5. (∇ 2 e X i Y j )p = 0 6. (∇ for all vector fields X, Y ∈ Γ(T M ) and p ∈ Γ(T 2 M ), where i, j = 1, 2 and (u, w) = S(p). 3. Biharmonicity Of section 3.1. The Curvature Tensor Definition 3.1. Let (M, g) be a Riemannian manifold and F : T M → T M be a smooth bundle endomorphism of T M . For λ = 0, 1, 2, the λ-lift of F to T 2 M is defined by F 0 = S −1 (HF, HF ), 1

F = F2 =

∗ S∗−1 (V F, 0), S∗−1 (0, V F ).

From Proposition 2.17, we obtain the following lemma. Lemma 3.2. Let F : T M → T M be a smooth bundle endomorphism of T M , then we have e X 1 F 0 )p = F (X)0 + 1 (R(u, X)F (u))0 , (∇ p p 2 1 e X 2 F 0 )p = F (X)0p + (R(w, X)F (w))0p , (∇ 2 e X i F j )p = F (X)jp i, j = 1, 2, (∇ e X 0 F 1 )p = (∇X F )1p + 1 (R(u, Fx (u))Xx )0 , (∇ 2 2 2 e X 0 F )p = (∇X F )p + 1 (R(w, Fx (w))Xx )0 , (∇ 2 0 0 e X 0 F )p = (∇X F ) − 1 (R(Xx , Fx (u))u)1 − 1 (R(Xx , Fx (w))w)2 (∇ p 2 2 for any p ∈ T 2 M,

i, j = 1, 2

and

X ∈ Γ(T M ).

Using the formula of curvature and Lemma 3.2, we have the following. e be curvature Proposition 3.3. Let R be a curvature tensor of (M, g), and R 2 D tensor of (T M, g ) equipped with the diagonal lift of g. Then we have the following 0 1 1 e 0 , Y 0 )Z 0 = 1. R(X R(X, Y )Z + R(u, R(Z, Y )u)X + R(w, R(Z, Y )w)X 4 4 1 0 1 + R(u, R(X, Z)u)Y + R(w, R(X, Z)w)Y 4 4 1 0 1 + R(u, R(X, Y )u)Z + R(w, R(X, Y )w)Z 2 2

172


e 0 , Y 0 )Z i 2. R(X

=

e 1 , Y 1 )Z 0 3. R(X

=

e i , Y 2 )Z 0 4. R(X

=

e i , Y 0 )Z 0 5. R(X

=

e i , Y 0 )Z j 6. R(X

=

e 1 , Y 2 )Z i 7. R(X

=

1 1 2 1 + ∇Z R)(X, Y )u + ∇Z R)(X, Y )w , 2 2 1 1 R(X, Y )Z + R(R(u, Z)Y, X)u + R(R(w, Z)Y, X)w 4 4 i 1 1 − R(R(u, Z)X, Y )u − R(R(w, Z)X, Y )w 4 4 1 (∇X R)(u, Z)Y + (∇X R)(w, Z)Y − (∇Y R)(u, Z)X + 2 0 − (∇Y R)(w, Z)X , 1 1 R(X, Y )Z + R(u, X)R(u, Y )Z + R(w, X)R(w, Y )Z 4 4 0 1 1 − R(u, Y )R(u, X)Z) − R(w, Y )R(w, X)Z) , 4 4 1 1 R(X, Y )Z + R(u, X)R(u, Y )Z + R(w, X)R(w, Y )Z 4 4 0 1 1 − R(u, Y )R(u, X)Z − R(w, Y )R(w, X)Z , 4 4 1 i 1 1 − R(u, Y )Z, X)u + R(w, Y )Z, X)w + R(X, Z)Y 4 4 2 0 1 + (∇X R)(u, Y )Z + (∇X R)(w, Y )Z , 2 1 0 1 1 R(Y, Z)X + R(u, Y )R(u, X)Z + R(w, Y )R(w, X)Z 2 4 4 e 1 , Y 1 )Z i = R(X e 2 , Y 2 )Z i = 0 R(X

for any ξ = (p, u, w) ∈ T 2 M,

i, j = 1, 2

and

X, Y, Z ∈ Γ(T M ).

Lemma 3.4. Let (M, g) be a Riemannian manifold and T 2 M be the tangent bundle equipped with the diagonal metric. If Z ∈ Γ(T M ) and σ ∈ Γ(T 2 M ), then (30)

b Z σ)Vp , dx σ(Zx ) = Zp0 + (∇

where p = σ(x). Proposition 3.5 ([3]). Let (M, g) be a Riemannian manifold and T 2 M be its tangent bundle of order two equipped with the diagonal metric. Then the tension field associated with σ ∈ Γ(T 2 M ) is τ (σ) = (traceg ∇2 Xσ )1 + (traceg ∇2 Yσ )2 0 + traceg (R(Xσ , ∇∗ Xσ ) ∗ +R(Yσ , ∇∗ Yσ )∗) (31) 0 b 2 σ)V + traceg (R(Xσ , ∇∗ Xσ ) ∗ +R(Yσ , ∇∗ Yσ )∗) , = (traceg ∇ b 2 ) denotes the Laplacian attached to ∇ where − traceg ∇2 (resp., − traceg ∇ b (resp., ∇).


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4. Biharmonicity of Section σ : (M, g) → (T 2 M, g D ) For a section σ ∈ Γ(T 2 M ), we denote (32)

τ 0 (σ) = τ 0 (Xσ ) + τ 0 (Yσ ),

(33)

τ V (σ) = τ 1 (Xσ ) + τ 2 (Yσ ), 0 τ 0 σ) = τ 0 (Xσ ) + τ 0 (Yσ ) , 1 2 τ V (σ) = τ 1 (Xσ ) + τ 2 (Yσ ) ,

(34) (35) where

τ 0 (Xσ ) = traceg (R(Xσ , ∇∗ Xσ )∗), τ 0 (Yσ ) = traceg (R(Yσ , ∇∗ Yσ )∗), τ 1 (Xσ ) = traceg ∇2 Xσ , τ 2 (Yσ ) = traceg ∇2 Yσ .

From these notations, we have (36)

τ (σ) = τ V + τ 0 .

Theorem 4.1. Let (M, g) be a Riemannian compact manifold and (T 2 M, g D ) be its tangent bundle of order two equipped with the diagonal metric and a vector bundle structure via the diffeomorphism S between T 2 and T M ⊕ T M . Then σ : M → T 2 M is a biharmonic section if and only if σ is harmonic. Proof. First, if σ is harmonic, then from Corollary 1.4, we deduce that σ is biharmonic. Conversely, assuming that σ is biharmonic. Let σt be a compactly supported variation of σ defined by σt = (1 + t)σ. Using Proposition 2.8, we have (37)

Xσt = (1 + t)Xσ

and Yσt = (1 + t)Yσ .

Substituting (37) in (32) to (35), we obtain (38)

τ 0 (σt ) = (1 + t)2 τ 0 (σ)

and τ V (σt ) = (1 + t)τ V (σ)

τ 0 (σt ) = (1 + t)2 τ 0 (σ) and τ V (σt ) = (1 + t)τ V (σ). Z Z Z Then 1 1 1 E2 (σt ) = |τ (σt )|2gD vg = |τ 0 (σt )|2gD vg + |τ V (σt )|2gD vg 2 2 2 Z Z (1 + t)4 (1 + t)2 |τ 0 (σ)|2gD vg + |τ V (σ)|2gD vg . = 2 2 Since the section σ is biharmonic, then for the variation σt , we have Z Z d 0 2 0 = E2 (σt )|t=0 = 2 |τ (σ)|gD vg + |τ V (σ)|2gD vg . dt Hence τ 0 (σ) = 0 and τ V (σ) = 0, then τ (σ) = 0. (39)

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In the case where M is not compact, the characterization of biharmonic sections requires the following two lemmas. Lemma 4.2. Let (M, g) be a Riemannian manifold and (T 2 M, g D ) be its tangent bundle of order two equipped with the diagonal metric. If σ ∈ Γ(T 2 M ) is a smooth section, then the Jacobi tensor Jσ (τ V (σ)) is given by n oV Jσ (τ V (σ)) = traceg ∇2 (τ V (σ)) n + traceg (R(u, ∇∗ τ 1 (Xσ ) ∗ +R(w, ∇∗ τ 2 (Yσ )) ∗ +R(τ V (σ), ∇∗ σ)∗ o0 1 1 + R(u, τ 1 (Xσ ))R(u, ∇∗ Xσ ) ∗ + R(w, τ 2 (Yσ ))R(w, ∇∗ Yσ )∗) . 2 2 Proof. Let p ∈ T 2 M and {ei }m i=1 be a local orthonormal frame on M such that (∇ei ei )x = 0. If we denote Fi (x, u, w) = 21 R(u, τ 1 (Xσ ))ei + 12 R(w, τ 2 (Yσ ))ei , then we have 1 1 2 2 e σ0 e σe τ V (σ)p = (∇ ∇ 2 (τ (Xσ )) + (τ (Yσ )) )p 1 i

ei +(∇ei Xσ ) +(∇ei Yσ )

1 = (∇ei (τ V (σ))Vp + (R(u, τ 1 (Xσ ))ei + R(w, τ 2 (Yσ ))ei )0 2 = (∇ei (τ V (σ))Vp + (Fi (x, u, w))0 , hence e 2 τ V (σ)p = (traceg ∇ = =

m n o X e σe ∇ e σe (τ V (σ)) ∇ i i

p

i=1 m n X

e σ0 ∇ e +(∇e i

i

Xσ

)1 +(∇

ei Yσ

)2

((∇ei (τ V (σ))V + (Fi )0

i=1 m n X

o p

e e0 (∇e τ 1 (Xσ ))1 + ∇ e e0 (∇e τ 1 (Yσ ))2 ∇ i i i i

i=1

e e0 F 0 + ∇ e (∇ X )1 F 0 + ∇ e (∇ Y )2 F 0 +∇ i i i σ ei σ ei i

o

.

p

Using Proposition 2.17, we obtain m n o1 X 1 e 2 τ V (σ))p = (traceg ∇ (∇ei ∇ei τ 1 (Xσ )) − R(ei , R(u, τ 1 (Xσ )ei )u 4 p i=1 +

m n m n o2 X X 1 1 (∇ei ∇ei τ 2 (Yσ )) − R(ei , R(w, τ 2 (Yσ )ei )w + R(u, ∇ei τ 1 (Xσ ))ei 4 2 p i=1 i=1

1 1 1 + R(w, ∇ei τ 2 (Yσ ))ei + (∇ei R(u, τ 1 (Xσ ))ei ) + (∇ei R(w, τ 2 (Yσ ))ei ) 2 2 2 1 1 1 + R(τ 1 (Xσ ), ∇ei u)ei + R(τ 2 (Yσ ), ∇ei w)ei + R(u, ∇ei Xσ )R(u, τ 1 (Xσ ))ei 2 2 4 o0 1 1 1 + R(w, ∇ei Yσ )R(w, τ 2 (Yσ ))ei + R(∇ei Xσ , τ 1 (Xσ ))ei + R(∇ei Xσ , τ 2 (Yσ ))ei . 4 2 2


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From proposition 3.3, we have e V (σ), dσ)dσ) traceg (R(τ =

m n X e 1 (Xσ ))1 , e0i )e0i + R((τ e 1 (Xσ ))1 , (∇e Xσ )1 )e0i R((τ i i=1

e 1 (Xσ ))1 , (∇e Yσ )2 )e0i + R((τ e 1 (Xσ ))1 , e0i )(∇e Xσ )1 + R((τ i i e 1 (Xσ ))1 , e0 )(∇e Yσ )2 + R((τ e 2 (Yσ ))2 , e0i )e0i + R((τ i i e 2 (Yσ ))2 , (∇e Xσ )1 )e0 + R((τ e 2 (Yσ ))2 , (∇e Yσ )2 )e0 + R((τ i i i i o e 2 (Yσ ))2 , e0i )(∇e Xσ )1 + R((τ e 2 (Yσ ))2 , e0i )(∇e Yσ )2 . + R((τ i i By calculating at point p ∈ T 2 M , we obtain e V (σ), dσ)dσ)p traceg (R(τ m n o1 n 1 o2 X 1 = − R(R(u, τ 1 (Xσ ))ei , ei )u − R(R(w, τ 2 (Yσ )ei ), ei )w 4 4 i=1 +

m n X R(τ 1 (Xσ ), ∇ei Xσ )ei + R(τ 2 (Yσ ), ∇ei Yσ )ei i=1

1 1 + R(u, τ 1 (Xσ ))R(u, ∇ei Xσ )ei − R(w, ∇ei Yσ )R(w, τ 2 (Yσ ))ei 4 4 1 1 + R(w, τ 2 (Yσ ))R(w, ∇ei Yσ )ei − R(u, ∇ei Xσ )R(u, τ 1 (Xσ ))ei 4 4 1 1 1 2 + R(τ (Xσ ), ∇ei Xσ )ei + R(τ (Yσ ), ∇ei Yσ )ei 2 2 1 1 1 + R(u, τ (Xσ )R(u, ∇ei Xσ )ei + R(u, τ 2 (Yσ )R(w, ∇ei Yσ )ei 4 4 o0 1 1 1 − (∇ei R(u, τ (Xσ )ei − (∇ei R(w, τ 2 (Yσ )ei . 2 2 Considering the formula (8), we deduce n oV n Jσ (τ V (σ)) = traceg ∇2 (τ V (σ)) + traceg (R(u, ∇∗ τ 1 (Xσ )∗ + R(w, ∇∗ τ 2 (Yσ )) ∗ +R(τ V (σ), ∇∗ σ)∗ o0 1 1 + R(u, τ 1 (Xσ ))R(u, ∇∗ Xσ ) ∗ + R(w, τ 2 (Yσ ))R(w, ∇∗ Yσ )∗) . 2 2 Lemma 4.3. Let (M, g) be a Riemannian manifold and (T 2 M, g D ) be its tangent bundle of order two equipped with the diagonal metric. If σ ∈ Γ(T 2 M ) is a

176


smooth section, then the Jacobi tensor Jσ (τ 0 (σ)) is given by Jσ (τ 0 (σ))p n o1 1 = traceg 2R(τ 0 (Xσ ), ∗)∇∗ Xσ −R(∗, ∇∗ τ 0 (Xσ ))u+ R(R(u, ∇∗ Xσ )∗, τ 0 (Xσ ))u 2 o2 n 1 0 0 + traceg 2R(τ (Yσ ), ∗)∇∗ Yσ − R(∗, ∇∗ τ (Yσ ))w + R(R(w, ∇∗ Yσ )∗, τ 0 (Yσ ))w 2 n + traceg ∇∗ ∇∗ τ 0 (σ) + R(u, ∇∗ Xσ )∇∗ τ 0 (Xσ ) + R(w, ∇∗ Yσ )∇∗ τ 0 (Yσ ) 1 1 R(u, ∇∗ ∇∗ Xσ )τ 0 (Xσ ) + R(w, ∇∗ ∇∗ Yσ )τ 0 (Yσ ) + R(u, R(τ 0 (Xσ ), ∗)u)∗ 2 2 + R(w, R(τ 0 (Yσ ), ∗)w) ∗ +R(τ 0 (σ), ∗) ∗ +(∇τ 0 (Xσ ) R)(u, ∇∗ Xσ )∗ o0 + (∇τ 0 (Yσ ) R)(w, ∇∗ Yσ ) ∗

+

p

2

for all p = (x, u, w) ∈ T M . Proof. Let p = (x, u, w) ∈ T 2 M and {ei }m i=1 be a local orthonormal frame on M such that (∇ei ei )x = 0, denoted by 1 1 R(ei , τ 0 (Xσ )) ∗ + R(ei , τ 0 (Yσ ))∗ 2 2 1 1 0 + GY = R(∗, ∇∗ Xσ )τ (Xσ ) + R(∗, ∇∗ Yσ )τ 0 (Yσ ). 2 2

(40)

Fi = FiX + FiY =

(41)

G = GX

First, using Lemma 3.4 and Proposition 2.17, we calculate e 2 (τ 0 (σ))p = traceg ∇

m n X

e σe ∇ e σe (τ 0 (σ))0 ∇ i i

o

i=1

=

m n X

e σ0 (∇ e +(∇e i

X )1+(∇ei Yσ )2 (∇ei τ (σ)) i σ

0

o 1 2 −FiX −FiY +G0i ) .

i=1

From Proposition 2.17, we have e 2 (τ 0 (σ))p = traceg ∇

m n X 1 (∇ei ∇ei τ 0 (σ))0 + ( R(u, ∇ei Xσ )∇ei τ 0 (Xσ ) 2 i=1

1 R(w, ∇ei Yσ )∇ei τ 0 (Yσ ))0 − (∇ei FiX )1 − (∇ei FiY )2 2 1 1 − ( R(ei , ∇ei τ 0 (Xσ ))u)1 − ( R(ei , ∇ei τ 0 (Yσ ))w)2 2 2 1 1 0 − (R(u, FiX (u))ei ) − (R(w, FiY (w))ei )0 2 2 − (FiX (∇ei )Xσ )1 − (FiY (∇ei )Yσ )2 + (∇ei G)0 1 1 − (R(ei , GX (u))u)1 − (R(ei , GY (w))w)2 2 2 + (GX (∇ei Xσ ))0 + (GY (∇ei Yσ ))0 o 1 1 + (R(u, ∇ei Xσ )GX (u))0 + (R(w, ∇ei Yσ )GY (w))0 . 2 2 p +

(42)

p


Substituting (40) and (41) in (42), we arrive at e 2 (τ 0 (σ))p traceg ∇ m n X = (∇ei ∇ei τ 0 (σ)) + R(u, ∇ei Xσ )∇ei τ 0 (Xσ ) i=1

1 R(u, ∇ei ∇ei Xσ )τ 0 (Xσ ) 2 1 1 R(w, ∇ei ∇ei Yσ )τ 0 (Yσ ) + (∇ei R)(u, ∇ei Xσ )τ 0 (Xσ ) 2 2 1 1 0 (∇ei R)(w, ∇ei Yσ )τ (Yσ ) + R(u, ∇ei Xσ )R(u, ∇ei Xσ )τ 0 (Xσ ) 2 4 1 1 R(w, ∇ei Yσ )R(w, ∇ei Yσ )τ 0 (Yσ ) − R(u, R(ei , τ 0 (Xσ ))u)ei 4 4 m n o0 X 1 1 0 R(w, R(ei , τ (Yσ ))w)ei − R(ei , τ 0 (Xσ ))∇ei Xσ 4 2 p i=1

+ R(w, ∇ei Yσ )∇ei τ 0 (Yσ ) + + + (43)

+ −

1 (∇ei R)(ei , τ 0 (Xσ ))u 2 o1 1 + R(ei , R(u, ∇ei Xσ )τ 0 (Xσ ))u 4 p m n X 1 − R(ei , τ 0 (Yσ ))∇ei Yσ + R(ei , ∇ei τ 0 (Yσ ))w 2 i=1 + R(ei , ∇ei τ 0 (Xσ ))u +

+

o2 1 1 (∇ei R)(ei , τ 0 (Yσ ))w + R(ei , R(w, ∇ei Yσ )τ 0 (Yσ ))w . 2 4 p

On the other hand, we have o n e 0 (σ), dσ)dσ traceg R(τ

p

m n X 3 = R(τ 0 (σ), ei )ei + R(u, R(τ 0 (Xσ ), ei )u)ei 4 i=1

3 R(w, R(τ 0 (Yσ ), ei )w)ei + ∇τ 0 (Xσ R)(u, ∇ei Xσ )ei 4 1 + ∇τ 0 (Yσ R)(w, ∇ei Yσ )ei − (∇ei R)(u, ∇ei Xσ )τ 0 (Xσ ) 2 1 1 − (∇ei R)(w, ∇ei Yσ )τ 0 (Yσ ) − R(u, ∇ei Xσ )R(u, ∇ei Xσ )τ 0 (Xσ ) 2 4 o0 1 0 − R(w, ∇ei Yσ )R(w, ∇ei Yσ )τ (Yσ ) 4 p m n X 1 1 0 (∇ei R)τ (Xσ , ei )u + R(R(u, ∇ei Xσ )ei , τ 0 (Xσ ))u + 2 2 i=1 o1 3 1 + R(τ 0 (Xσ ), ei )∇ei Xσ − R(R(u, ∇ei Xσ )τ 0 (Xσ ), ei )u 2 4 m n X 1 1 0 (∇ei R)τ (Yσ , ei )w + R(R(w, ∇ei Yσ )ei , τ 0 (Yσ ))w + 2 2 i=1 o2 3 1 + R(τ 0 (Yσ ), ei )∇ei Yσ − R(R(w, ∇ei Yσ )τ 0 (Yσ ), ei )w . 2 4 +

(44)

177

178


By summing (43) and (44), the proof of Lemma 4.3 is completed.

From Lemma 4.2 and 4.3, we deduce the following theorems Theorem 4.4. Let (M, g) be a Riemannian manifold and (T 2 M, g D ) be its tangent bundle of order two equipped with the diagonal metric. If σ : M → T 2 M is a smooth section, then the bitension field of σ is given by n τ2 (σ)p = traceg ∇2 τ 1 (Xσ ) + 2R(τ 0 (Xσ ), ∗)∇∗ Xσ − R(∗, ∇∗ τ 0 (Xσ ))u o1 1 + R(R(u, ∇∗ )∗, τ 0 (Xσ ))u 2 n + traceg ∇2 τ 2 (Yσ ) + 2R(τ 0 (Yσ ), ∗)∇∗ Yσ o2 1 − R(∗, ∇∗ τ 0 (Yσ ))w + R(R(w, ∇∗ )∗, τ 0 (Yσ ))w 2 n 1 + traceg R(u, ∇∗ τ (Xσ )) ∗ +R(w, ∇∗ τ 2 (Yσ ))∗ + R(τ 1 (Xσ ), ∇∗ Xσ )∗ 1 + R(τ 2 (Yσ ), ∇∗ Yσ ) ∗ + R(u, τ 1 (Xσ ))R(u, ∇∗ Xσ )∗ 2 1 2 + R(w, τ (Yσ ))R(w, ∇∗ Yσ ) ∗ +∇∗ ∇∗ τ 0 (σ) 2 + R(u, ∇∗ Xσ )∇∗ τ 0 (Xσ ) + R(w, ∇∗ Yσ )∇∗ τ 0 (Yσ ) + R(τ 0 (σ), ∗)∗ 1 + R(u, ∇∗ ∇∗ Xσ )τ 0 (Xσ ) 2 1 + R(w, ∇∗ ∇∗ Yσ )τ 0 (Yσ ) + R(u, R(τ 0 (Xσ ), ∗)u)∗ 2 + R(w, R(τ 0 (Yσ ), ∗)w) ∗ +(∇τ 0 (Xσ ) R)(u, ∇∗ Xσ )∗ o0 + (∇τ 0 (Yσ ) R)(w, ∇∗ Yσ ) ∗ p

for all p ∈ T 2 M . Theorem 4.5. Let (M, g) be a Riemannian manifold and (T 2 M, g D ) be its tangent bundle of order two equipped with the diagonal metric. A section σ : M → T 2 M is biharmonic ifnand only if the following conditions are verified: 1) 0 = traceg ∇2 τ 1 (Xσ ) + 2R(τ 0 (Xσ ), ∗)∇∗ Xσ − R(∗, ∇∗ τ 0 (Xσ ))u o 1 + R(R(u, ∇∗ )∗, τ 0 (Xσ ))u , p n 2 2)

0 = traceg ∇2 τ 2 (Yσ ) + 2R(τ 0 (Yσ ), ∗)∇∗ Yσ − R(∗, ∇∗ τ 0 (Yσ ))w o + 21 R(R(w, ∇∗ )∗, τ 0 (Yσ ))w , p


3)

179

n 0 = traceg R(u, ∇∗ τ 1 (Xσ )) ∗ +R(w, ∇∗ τ 2 (Yσ ))∗ + R(τ 1 (Xσ ), ∇∗ Xσ ) ∗ +R(τ 2 (Yσ ), ∇∗ Yσ )∗ 1 + R(u, τ 1 (Xσ ))R(u, ∇∗ Xσ )∗+ 21 R(w, τ 2 (Yσ ))R(w, ∇∗ Yσ )∗ 2 + ∇∗ ∇∗ τ 0 (σ) + R(u, ∇∗ Xσ )∇∗ τ 0 (Xσ ) + R(w, ∇∗ Yσ )∇∗ τ 0 (Yσ ) 1 + R(u, ∇∗ ∇∗ Xσ )τ 0 (Xσ ) + 21 R(w, ∇∗ ∇∗ Yσ )τ 0 (Yσ ) 2 + R(u, R(τ 0 (Xσ ), ∗)u) ∗ +R(w, R(τ 0 (Yσ ), ∗)w)∗ + R(τ 0 (σ), ∗) ∗ +(∇τ 0 (Xσ ) R)(u, ∇∗ Xσ )∗ o + (∇τ 0 (Yσ ) R)(w, ∇∗ Yσ ) ∗ p

for all p = S −1 (x, u, w) ∈ T 2 M . Corollary 4.6. Let (M, g) be a Riemannian manifold and (T 2 M, g D ) be its tangent bundle of order two equipped with the diagonal metric. If σ : M → T 2 M is a section such that Xσ and Yσ are biharmonic vector fields, then σ is biharmonic. (For biharmonic vector see [1]). References 1. Djaa M., EL Hendi H. and Ouakkas S., Biharmonic vector field, Turkish J. Math. 36 (2012), 463–474. 2. Djaa M. and Gancarzewicz J., The geometry of tangent bundles of order r, Boletin Academia, Galega de Ciencias, Espagne, 4 (1985), 147–165. 3. Djaa N. E. H., Ouakkas S. and Djaa M., Harmonic sections on the tangent bundle of order two, Ann. Math. Inform. 38 ( 2011), 15–25. 4. Eells J. and Sampson J.H., Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–60. 5. Ishihara T., Harmonic sections of tangent bundles, J. Math. Univ. Tokushima 13 (1979), 23–27. 6. Jiang G. Y., Harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A 7 (1986) 389–402. 7. Konderak J. J., On Harmonic Vector Fields, Publications Matmatiques 36 (1992), 217–288. 8. Miron R., The Geometry of Higher Order Hamilton Spaces, arXiv: 1003.2501 [mathDG], 2010. 9. Oproiu V., On Harmonic Maps Between Tangent Bundles, Rend. Sem. Mat, 47 (1989), 47–55. H. Elhendi, Department of Mathematics University of Bechar, 08000 Bechar, Algeria., current address: Cit´ e Aissat Idir Rue BenHaboucha No 27, Relizane 48000 Algerie, e-mail: [email protected] M. Terbeche, LGACA Laboratory. Department of Mathematics, Oran Es-Senia University, 310000, Algeria., e-mail: [email protected] M. Djaa, GMFAMI Laboratory. Department of Mathematics Center University of Relizane, 48000, Algeria., e-mail: [email protected]