Harmonic sections on the tangent bundle of order two

38(2011) pp. 15–25

http://ami.ektf.hu

Harmonic sections on the tangent bundle of order two ^∗

Nour El Houda Djaa, Seddik Ouakkas, Djaa Mustapha

Laboratory of Geometry, Analysis, Control and Applications Department of Mathematics, University of Saida, Saida, Algeria.

e-mail: Lgaca_saida2009@hotmail.com

Submitted July 25, 2010 Accepted May 30, 2011

Abstract

The problem studied in this paper is related to the Harmonicity of sections from a Riemannian manifold(M, g)into its tangent bundle of order twoT²M equipped with the Diagonal metricg^D. First we introduce a connection on Γ(T²M)and we investigate the geometry and the harmonicity of sections as maps from(M, g)to(T²M, g^D).

Keywords:Horizontal lift, vertical lift, harmonic maps.

MSC:53A45, 53C20, 58E20

1. Introduction

Consider a smooth mapφ: (M^m, g)→(Nⁿ, h)between two Riemannian manifolds, then the energy functional is defined by

E(φ) =1 2

Z

M

|dφ|²dvg (1.1)

(or over any compact subsetK⊂M).

A map is called harmonic if it is a critical point of the energy functionalE (or E(K)for all compact subsetsK⊂M). For any smooth variation{φ}_t∈I ofφwith φ0=φandV = ^dφ_dt^t

_t=0, we have d

dtE(φt) _t=0

=− Z

M

h(τ(φ), V)dvg, (1.2)

∗The authors would like to thank the referee for his useful remarks.

15

where

τ(φ) = traceg∇dφ (1.3)

is the tension field ofφ. Then we have

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

τ(φ) = 0. (1.4)

If(xⁱ)_1≤i≤mand(y^α)_1≤α≤ndenote local coordinates onM andN respectively, then equation (1.4) takes the form

τ(φ)^α=

∆φ^α+g^{ij N}Γ^α_βγ∂φ^β

∂xⁱ

∂φ^γ

∂x^j

= 0, 1≤α≤n, (1.5) where∆φ^α=√¹

|g|

∂

∂xⁱ

p

|g|g^{ij ∂φ}_∂x^αj

is the Laplace operator on(M^m, g)and^NΓ^α_βγ are the Christoffel symbols onN. One can refer to [1, 4, 6, 7, 8, 9] for background on harmonic maps.

2. Some results on horizontal and vertical lifts

Let(M, g)be an n-dimensional Riemannian manifold and(T M, π, M)be its tan- gent 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 by Γ^k_ij the Christoffel symbols ofg and by∇the Levi-Civita connection ofg.

We have two complementary distributions onT M, the vertical distributionV and the horizontal distribution H, defined by:

V_(x,u)=Ker(dπ_(x,u))

={aⁱ ∂

∂yⁱ|(x,u); aⁱ∈R} H_(x,u)={aⁱ ∂

∂xⁱ|_(x,u)−aⁱu^jΓ^k_ij ∂

∂y^k|_(x,u); aⁱ∈R}, where(x, u)∈T M, such thatT_(x,u)T M =H(x,u)⊕ V(x,u).

LetX =X^{i ∂}_∂xi be a local vector field onM. The vertical and the horizontal lifts ofX are defined by

X^V =Xⁱ ∂

∂yⁱ (2.1)

X^H =Xⁱ δ

δxⁱ =Xⁱ{ ∂

∂xⁱ −y^jΓ^k_ij ∂

∂y^k} (2.2)

For consequences, we have(_∂x^∂_i)^H = _δx^δ_i,(_∂x^∂_i)^V = _∂y^∂_i and(_δx^δ_i,_∂y^∂_j)i,j=1,...,n

a local frame on T M.

Remark 2.1.

1. ifw=w^{i ∂}_∂xi +w^{j ∂}_∂yj ∈T(x,u)T M, then its horizontal and vertical parts are defined by

w^h=wⁱ ∂

∂xⁱ −wⁱu^jΓ^k_ij ∂

∂y^k ∈ H_(x,u) w^v ={w^k+wⁱu^jΓ^k_ij} ∂

∂y^k ∈ V(x,u)

2. ifu=u^{i ∂}_∂xi ∈T_xM then its vertical and horizontal lifts are defined by u^V =uⁱ ∂

∂yⁱ u^H =uⁱ{ ∂

∂xⁱ −y^jΓ^k_ij ∂

∂y^k}.

Proposition 2.2(see [10]). LetF ∈T¹_p(M)be a tensor of type (1,p) (respectively, G ∈T⁰_p(M) a tensor of type (0,p)), then there exist a tensor γ(F) ∈T¹_p−1(T M) (respectively, γ(G)∈T⁰_p−1(T M)), localy defined by

γ(F) =F_h^k

1h₂...h_py^h1 ∂

∂y^k ⊗dx^h2⊗ · · · ⊗dx^hp (2.3) γ(G) =Gh₁h₂...h_py^h1dx^h2⊗ · · · ⊗dx^hp (2.4) whereF =F_i^j

1...i_p

∂

∂x^j ⊗dxⁱ¹⊗ · · · ⊗dx^ip andG=Gi₁...i_pdxⁱ¹⊗ · · · ⊗dx^ip. Definition 2.3. The Sasaki metric g^s on the tangent bundle T M ofM is given by

1. g^s(X^H, Y^H) =g(X, Y)◦π 2. g^s(X^H, Y^V) = 0

3. g^s(X^V, Y^V) =g(X, Y)◦π for all vector fieldsX, Y ∈Γ(T M).

In the general case, Sasaki metrics is considered in [2, 5, 7, 10].

Proposition 2.4 (see [6]). A vector fieldsX : (M, g)→(T M, g^s) is harmonic iff X

i=1

X_ii^k = 0, X

i=1

R^k_iljX_i^j= 0.

whereX_i^k (respX_ij^k) are the components of the first (resp second) covariant differ- ential of the vector fieldX.

From Proposition 2.4 we deduce

Proposition 2.5. If X : (M, g)→(T M, g^s)is a harmonic vector field, then trace_g∇²X= 0, trace_gR(X,∇∗X)∗= 0.

Let M be an n-dimensional manifold. The tangent bundle of order 2 is the natural bundle of 2-jets of differentiable curves, defined by:

T²M ={j₀²γ ; γ:R0→M, is a smooth curve at 0∈R} π₂:T²M →M

j₀²γ7→γ(0)

A local chart(U, xⁱ)i=1...n onM induces a local chart (π₂⁻¹(U), xⁱ, yⁱ, zⁱ)i=1...n

onT²M by the following formulae xⁱ=γⁱ(0).

yⁱ=_dt^dγⁱ(0).

zⁱ= _dt^d22γⁱ(0).

Proposition 2.6. Let M, be an n-dimensional manifold, then T M is sub-bundle of T²M, and the map

i:T M →T²M

j¹₀f =j₀²fe (2.5)

is an injective homomorphism of a natural bundles (not of vector bundles), where

feⁱ= Z t

0

fⁱ(s)ds−tfⁱ(0) +fⁱ(0) i= 1. . . n.

Proof. Locally if (U, xⁱ) is a chart on M and (U, xⁱ, yⁱ) and (U, xⁱ, yⁱ, zⁱ)are the induced chart onT M andT²M respectivelly, then we havei: (xⁱ, yⁱ)7→(xⁱ,0, yⁱ), it follows that i is an injective homomorphism. Remains to show that i is well defined.

Let(U, ϕ)and(V, ψ)are a charts on M, for any vector j₀¹f ∈T M, if we denote

fe(t) =ϕ⁻¹( Z t

0

ϕ◦f(s)ds−tϕ◦f(0) +ϕ◦f(0))

fb(t) =ψ⁻¹( Z t

0

ψ◦f(s)ds−tψ◦f(0) +ψ◦f(0)) then we obtain

ϕ◦fe(0) =ϕ◦f(0)

=ϕ◦fb(0) d

dt(ϕ◦fe)(0) = 0

= d

dt(ϕ◦fb)(0) d²

dt²(ϕ◦fe)(0) = d

dt(ϕ◦f)(0)

= d²

dt²(ϕ◦fb)(0) which proves that j₀²fe=j₀²fb.

Theorem 2.7. Let (M, g) be a Riemannian manifold and ∇ be the Levi-Civita connection. If T M⊕T M denotes the Whitney sum, then

S:T²M →T M⊕T M

j₀²γ7→( ˙γ(0),(∇γ(0)˙ γ)(0))˙ (2.6) is a diffeomorphism of natural bundles.

In the induced coordinate, we have

S: (xⁱ, yⁱ, zⁱ)7→(xⁱ, yⁱ, zⁱ+y^jy^kΓⁱ_jk) (2.7) In the more general case, the difeomorphismS is considered in [3].

Remark 2.8. The diffeomorphismSdetermines a vector bundle structure onT²M, for which S be an isomorphism of vector bundles, and i : T M → T²M is an injective homomorphism of vector bundles.

Definition 2.9. Let(M, g) be a Riemannian manifold and T²M be its tangent bundle of order 2 endowed with the vectorial structure induced by the diffeomor- phismS. For any sectionσ∈Γ(T²M), we define two vector fields onM by:

X_σ=P₁◦S◦σ

Yσ=P2◦S◦σ (2.8)

where P1 and P2 denotes the first and the second projection from T M⊕T M on T M.

Remark 2.10. We can easily verify that for all sectionsσ, $∈Γ(T²M)andα∈R, we have

X_ασ+$=αX_σ+X_$ Yασ+$=αYσ+Y$

From the Remarks 2.8 and 2.10 we can define a connection onΓ(T²M).

Definition 2.11. Let(M, g)be a Riemannian manifold and T²M be its tangent bundle of order 2 endowed with the vectorial structure induced by the diffeomor- phismS. We define a connection onΓ(T²M)by:

∇b : Γ(T M)×Γ(T²M)→Γ(T²M)

(Z, σ)7→∇bZσ=S⁻¹((∇ZX_σ,∇ZY_σ)) (2.9) where∇ is the Levi-Civita connection onM.

From formula 2.7 and Definition 2.9 , it follows

Proposition 2.12. If (U, xⁱ) is a chart on M and (σⁱ, σⁱ)are the components of sectionσ∈Γ(T²M) then

X_σ =σⁱ ∂

∂xⁱ

Y_σ = (σ^k+σⁱσ^jΓ^k_ij) ∂

∂x^k

From Theorem 2.7 and Remark 2.10 we have

Proposition 2.13. Let(M, g)be a Riemannian manifold andT²M be its tangent bundle of order 2, then

J : Γ(T M)→Γ(T²M)

Z=S⁻¹(Z,0) (2.10)

is an injective homomorphism of vector bundles.

Locally if(U, xⁱ)is a chart onM and(U, xⁱ, yⁱ)and(U, xⁱ, yⁱ, zⁱ)are the induced chart on T M andT²M respectivelly, then we have

J : (xⁱ, yⁱ)7→(xⁱ, yⁱ,−y^jy^kΓⁱ_jk) (2.11) Definition 2.14. Let (M, g) be a Riemannian manifold and X ∈ Γ(T M) be a vector field onM. Forλ= 0,1,2, theλ-lift ofX toT²M is defined by

X⁰=S_∗⁻¹(X^H, X^H) X¹=S_∗⁻¹(X^V,0)

X²=S_∗⁻¹(0, X^V) (2.12)

In the more general case, theλ-lift is considered in [3].

Theorem 2.15 (see [3]). Let (M, g)be a Riemannian manifold and R its tensor curvature, then for all vector fields X, Y ∈Γ(T M)andp∈T²M we have

1. [X⁰, Y⁰]_p= [X, Y]⁰_p−(R(X, Y)u)¹−(R(X, Y)w)²

2. [X⁰, Yⁱ] = (∇XY)ⁱ 3. [Xⁱ, Y^j] = 0.

where(u, w) =S(p)andi, j= 1,2.

Definition 2.16. Let (M, g) be a Riemannian manifold. For any section σ ∈ Γ(T²M)we define the vertical lift ofσtoT²M by

σ^V =S_∗⁻¹(X_σ^V, Y_σ^V)∈Γ(T(T²M)). (2.13) Remark 2.17. From Definition 2.9 and the formulae (2.5), (2.10), (2.12) and (2.13), for allσ∈Γ(T²M)andZ∈Γ(T M), we obtain

• σ^V =X_σ¹+Y_σ²

• (∇bZσ)^V = (∇ZXσ)¹+ (∇ZYσ)²

• Z¹=J(Z)^V

• Z²=i(Z)^V

3. Metric diagonal and harmonicity

Using Definition 2.3 and formula (2.12), we have

Theorem 3.1. Let(M, g)be a Riemannian manifold andT M its tangent bundle equipped with the Sasakian metric g^s, then

g^D=S_∗⁻¹(eg⊕eg) is the only metric that satisfies the following formulae

g^D(Xⁱ, Y^j) =δij·g(X, Y)◦π2 (3.1) for all vector fieldsX, Y ∈Γ(T M)andi, j= 0, . . . ,2, whereegis the metric defined by

eg(X^H, Y^H) =1

2g^s(X^H, Y^H) eg(X^H, Y^V) =g^s(X^H, Y^V) eg(X^V, Y^V) =g^s(X^V, Y^V), g^D is called the diagonal lift ofgto T²M.

Theorem 3.2. Let (M, g) be a Riemannian manifold and ∇e be the Levi-Civita connection of the tangent bundle of order two T²M equipped with the diagonal metricg^D. Then

1. (∇eX⁰Y⁰)p= (∇XY)⁰−¹₂(R(X, Y)u)¹−¹₂(R(X, Y)w)², 2. (∇eX⁰Y¹)p= (∇XY)¹+¹₂(R(u, Y)X)⁰,

3. (∇e_X0Y²)p= (∇XY)²+¹₂(R(w, Y)X)⁰, 4. (∇e_X1Y⁰)p=¹₂(Rx(u, X)Y))⁰,

5. (∇e_X2Y⁰)p=¹₂(Rx(w, X)Y))⁰, 6. (∇e_XiY^j)p= 0

for all vector fieldsX, Y ∈Γ(T M)andp∈Γ(T²M), where i, j= 1,2 and(u, w) = S(p).

The proof of theorem 3.2 follows directly from Theorem 3.1 and the Kozul formula.

Lemma 3.3. Let (M, g) be a Riemannian manifold and(T M, g^s) be the tangent bundle equipped with the Sasaki metric. If X, Y ∈Γ(T M) are a vector fields and (x, u)∈T M such that X_x=u, then we have

dxX(Yx) =Y_(x,u)^H + (∇YX)^V_(x,u).

Proof. Let (U, xⁱ) be a local chart on M in x ∈ M and (π⁻¹(U), xⁱ, y^j) be the induced chart onT M, ifXx=Xⁱ(x)_∂x^∂i|xandYx=Yⁱ(x)_∂x^∂i|x, then

dxX(Yx) =Yⁱ(x) ∂

∂xⁱ|_(x,Xx)+Yⁱ(x)∂X^k

∂xⁱ (x) ∂

∂y^k|_(x,Xx), thus the horizontal part is given by

(d_xX(Y_x))^h=Yⁱ(x) ∂

∂xⁱ|_(x,Xx)−Yⁱ(x)X^j(x)Γ^k_ij(x) ∂

∂y^k|_(x,Xx)

=Y_(x,X^H _x) and the vertical part is given by

(d_xX(Y_x))^v={Yⁱ(x)∂X^k

∂xⁱ (x) +Yⁱ(x)X^j(x)Γ^k_ij(x)} ∂

∂y^k|_(x,Xx)

= (∇YX)^V_(x,Xx).

Lemma 3.4. Let(M, g)be a Riemannian manifold and(T²M, g^D)be the tangent bundle equipped with the diagonal metric. If Z ∈Γ(T M) andσ∈Γ(T²M) , then we have

dxσ(Zx) =Z_p⁰+ (b∇Zσ)^V_p. (3.2) wherep=σ(x).

Proof. Using Lemma 3.3, we obtain

d_xσ(Z) =dS⁻¹(dX_σ(Z), dY_σ(Z))_S(p)

=dS⁻¹(Z^h, Z^h)_S(p)+dS⁻¹((∇ZXσ)^v,(∇ZYσ)^v)_S(p)

=Z_p⁰+ (∇bZσ)^V_p.

Lemma 3.5. Let (M, g)be a Riemannian n-dimensional manifold and(T²M, g^D) be its tangent bundle of order two equipped with the diagonal metric and let σ ∈ Γ(T²M). Then the energy density associated withσis

e(σ) =n 2 +1

2k∇σkb ². wherek∇σkb ²= trace_gg(∇Xσ,∇Xσ) + trace_gg(∇Yσ,∇Yσ).

Proof. Let(e₁, . . . , e_n)be a local orthonormal frame on M, then e(σ) =1

2

n

X

i=1

g^D(dσ(ei), dσ(ei))

Using formula 3.2 and Remark 2.17, we obtain

e(σ) = 1 2

n

X

i=1

g^D(e⁰_i, e⁰_i) +1 2

n

X

i=1

g^D((∇beiσ)^V,(∇beiσ)^V)

= n 2 +1

2k∇σkb ².

Theorem 3.6. Let(M, g)be a Riemannian manifold and(T²M, g^D)be its tangent bundle of order two equipped with the diagonal metric. Then the tension field associated with σ∈Γ(T²M)is

τ(σ) = (trace_g∇b²σ)^V + (trace_g{R(Xσ,∇∗X_σ)∗+R(Y_σ,∇∗Y_σ)∗})⁰. (3.3) Proof. Let x ∈ M and {e_i}ⁿ_i=1 be a local orthonormal frame on M such that

∇e_iej= 0, then

τ(σ)_x=

n

X

i=1

(∇dσ(e_i)dσ(e_i))_σ(x)

=

n

X

i=1

h∇_e0

i+(∇_eiσ)^V

e⁰_i + (∇b_eiσ)^Vi

σ(x)

From Theorem 3.2, we obtain

τ(σ)x=

n

X

i=1

n∇_e0

ie⁰_i +∇_e0

i(∇e_iXσ)¹+∇_e0

i(∇e_iYσ)²+∇_(∇eiXσ)1e⁰_i +∇_(∇eiYσ)2e⁰_io

σ(x)

=

n

X

i=1

n(∇e_i∇e_iXσ)¹_σ(x)+ (∇e_i∇e_iYσ)²_σ(x)+ (Rx(Xσ(x),∇e_iXσ)ei)⁰

+ (Rx(Yσ(x),∇e_iYσ)ei)⁰o

Theorem 3.7. Let(M, g)be a Riemannian manifold and(T²M, g^D)be its tangent bundle of order two equipped with the diagonal metric. A section σ:M →T²M is harmonic if and only the following conditions are verified

trace_g(∇²X_σ) = 0, traceg(∇²Yσ) = 0,

traceg{R(Xσ,∇_∗Xσ)∗+R(Yσ,∇_∗Yσ)∗}= 0.

From Proposition 2.5 and Theorem 3.7 we obtain

Corollary 3.8. Let(M, g)be a Riemannian manifold and(T²M, g^D)be its tangent bundle of order two equipped with the diagonal metric. Ifσ:M →T²M is a section such that Xσ andYσ are harmonic vector fields, thenσis harmonic.

Corollary 3.9. Let(M, g)be a Riemannian manifold and(T²M, g^D)be its tangent bundle of order two equipped with the diagonal metric. Ifσ:M →T²M is a section such that Xσ andYσ are parallel, thenσis harmonic.

Theorem 3.10. Let(M, g)be a Riemannian compact manifold and(T²M, g^D)be its tangent bundle of order two equipped with the diagonal metric. Then σ:M → T²M is a harmonic section if and only ifσ is parallel (i.e∇σb = 0).

Proof. Ifσis parallel, from Corollary 3.9, we deduce thatσis harmonic. Inversely.

Let σt be a compactly supported variation ofσ defined by σt = (1 +t)σ. From Lemma 3.5 we have

e(σt) = n

2 +(t+ 1)² 2 k∇σkb ². Ifσis a critical point of the energy functional we have :

0 = d

dtE(σt)_|t=0,

= Z

M

k∇σkb ²dv_gD

Hence ∇σb = 0.

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