Annales Mathematicae et Informaticae (38.)

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TOMUS 38. (2011)

COMMISSIO REDACTORIUM

Sándor Bácsó (Debrecen), Sonja Gorjanc (Zagreb), Tibor Gyimóthy (Szeged), Miklós Hoffmann (Eger), József Holovács (Eger), László Kozma (Budapest), Kálmán Liptai (Eger), Florian Luca (Mexico), Giuseppe Mastroianni (Potenza), Ferenc Mátyás (Eger), Ákos Pintér (Debrecen), Miklós Rontó (Miskolc, Eger), László Szalay (Sopron), János Sztrik (Debrecen, Eger), Gary Walsh (Ottawa)

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International journal for mathematics and computer science Referred by

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ANNALES

MATHEMATICAE ET INFORMATICAE

VOLUME 38. (2011)

EDITORIAL BOARD

Sándor Bácsó (Debrecen), Sonja Gorjanc (Zagreb), Tibor Gyimóthy (Szeged), Miklós Hoffmann (Eger), József Holovács (Eger), László Kozma (Budapest), Kálmán Liptai (Eger), Florian Luca (Mexico), Giuseppe Mastroianni (Potenza), Ferenc Mátyás (Eger), Ákos Pintér (Debrecen), Miklós Rontó (Miskolc, Eger), László Szalay (Sopron), János Sztrik (Debrecen, Eger), Gary Walsh (Ottawa)

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HU ISSN 1787-6117 (Online)

A kiadásért felelős az Eszterházy Károly Főiskola rektora Megjelent az EKF Líceum Kiadó gondozásában

Kiadóvezető: Kis-Tóth Lajos Felelős szerkesztő: Zimányi Árpád Műszaki szerkesztő: Tómács Tibor Megjelent: 2011. december Példányszám: 30

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Annales Mathematicae et Informaticae 38(2011) pp. 3–13

http://ami.ektf.hu

On general strong laws of large numbers for fields of random variables

Cheikhna Hamallah Ndiaye

^a

, Gane Samb LO

^b

aLERSTAD, Université de Saint-Louis LMA - Laboratoire de Mathématiques Appliquées Université Cheikh Anta Diop BP 5005 Dakar-Fann Sénégal

e-mail: chndiaye@ufrsat.org

bLaboratoire de Statistiques Théoriques et Appliquées (LSTA) Université Pierre et Marie Curie (UPMC) France

LERSTAD, Université de Saint-Louis e-mail:ganesamblo@ufrsat.org

Submitted September 6, 2011 Accepted November 25, 2011

Abstract

A general method to prove strong laws of large numbers for random fields is given. It is based on the Hájek-Rényi type method presented in Noszály and Tómács [14] and in Tómács and Líbor [16]. Noszály and Tómács [14]

obtained a Hájek-Rényi type maximal inequality for random fields using moments inequalities. Recently, Tómács and Líbor [16] obtained a Hájek-Rényi type maximal inequality for random sequences based on probabilities, but not for random fields. In this paper we present a Hájek-Rényi type maximal inequality for random fields, using probabilities, which is an extension of the main results of Noszály and Tómács [14] by replacing moments by probabilities and a generalization of the main results of Tómács and Líbor [16]

for random sequences to random fields. We apply our results to establish- ing a logarithmically weighted sums without moment assumptions and under general dependence conditions for random fields.

Keywords: Strong laws of large numbers, Maximal inequality, Probability inequalities, Random fields.

MSC:Primary 60F15 60G60. Secondary 62H11 62H35.

3

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1. Introduction and notations

We are concerned in this paper with strong laws of large numbers (SLLN) for random fields. Although this type of problems is entirely settled for sequence of independent real random variables (see for instance [9]) and general strong laws of large numbers for dependent real random variables based on Hájek-Rényi types inequalities. But as for random fields, they are still open. As a reminder, we recall that a family of random elements (X_n)_n∈T is said to be a random field if the set is endowed with a partial order (≤), not necessarily complete. For example, and it is the case in this paper,T may beN^d, whered >1 is an integer andN is the set of nonnegative integers. For such a real random field (Xn)_n∈Nd, we intend to contribute to assessing the more general SLLN’s, that is finding general conditions under which there exists a real number µ and a family of normalizing positive numbers (bn)_n∈Nd, named here as a d-sequence, such that, for S_(0,...,0) = 0, and Sn =P

m≤nXm forn >0, one has

Sn/bn→µ, a.s.

In the case of random fields, the data may be heavily dependent and then Hájek- Rényi type maximal inequalities are needed to obtain strong laws of large numbers, like in the real case. It seems that providing such inequalities goes back to Móricz [11] and Klesov [8]. Based on such inequalities, many authors established strong laws of large numbers such as Nguyen et al. [13], Tómács [19], Lagodowski [10], Noszály and Tómács [14], Móricz [12], Klesov [8], Fazekas et al. [5], Fazekas [2], [4]

and the literature cited herein.

One of the motivations of finding general strong laws of large numbers comes from that the finding, as proved by Cairoli [1], that classical maximal probability inequalities for random sequences are not valid in general for random fields.

Besides, nonparametric estimation for random fields or spatial processes was given increasing and simulated attention over the last few years as a consequence of grow- ing demands from applied research areas (see for instance Guyon [6]). This results in the serious motivation to extend the Hájek-Rényi type maximal inequality for probabilities for random sequences, what the cited above authors tackled.

Our objective is to give a nontrivial generalization of some fundamental results of these authors that will lead to positive answers to classical and non solved SLLN’s. Before a more precise formulation of our problem, we need a few additional notation.

From now on d is a fixed positive integer. The elements of N^d will be written in font bold like n while their coordinate are written in the usual way like n = (n1, . . . , nd). N^d is endowed with the usual partial ordering, that is n = (n1, . . . , nd) ≤ m = (m1, . . . , nd) if and only if or each 1 ≤ i ≤ d, one has ni ≤ mi. Further m < n means m ≤ n and n 6= m. We specially denote (1, . . . ,1)≡1and(0, . . . ,0)≡0. All the limits, unless specification, are meant as n= (n₁, . . . , n_d)→ ∞, that is equivalent to say thatn_i → ∞for each1≤i≤d.

To finish, any family of real numbers (b_n)_n∈A indexed by a subsetN^d is called a

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On general strong laws of large numbers for fields of random variables 5 d-sequence. We intensively use product typed-sequences. Ad-sequence(bn)n∈Ais of product type if it may be written in the form

b_n= Y

1≤i≤d

b⁽ⁱ⁾_n

i.

This product type d-sequence is unbounded and nondecreasing if and only if each sequence b⁽ⁱ⁾n_i is unbounded and nondecreasing in n_i. Now with these minimum notation, we are able to state the results of Tómács, Líbor and their co-authors.

On one hand, it is known that the Hájek-Rényi type maximal inequality (see [3]) is an important tool for proving SLLN’s for sequences. It is natural that Noszály and Tómács [14] used a generalization of this result for random fields in order to get SLLN’s for such objects. They stated

Proposition 1.1. Letrbe a positive real number,anbe a nonnegatived-sequence.

Suppose that bn is a positive, nondecreasingd-sequence of product type. Then E(max

`≤n|S`|^r)≤X

`≤n

a` ∀n∈N^d implies

E

max`≤n|S`|^rb^−r_`

≤4^dX

`≤n

a`b^−r_` ∀n∈N^d.

From this, they were led to the following general SLLN for random fields.

Theorem 1.2. Let an, bn be non-negative d-sequences and let r > 0. Suppose that bn is a positive, nondecreasing, unbounded d-sequence of product type. Let us assume that

X

n

an

b^r_n <∞ and

E

maxm≤n|Sm|^r

≤ X

m≤n

am ∀n∈N^d. Then

n→∞lim Sn

bn

= 0 a.s.

On an other hand, Tómács and Líbor [16], introduced a Hájek-Rényi inequality for probabilities and, subsequently, strong laws of large numbers for random sequences but not for random fields. They obtained first:

Theorem 1.3. Let r be a positive real number, an be a sequence of nonnegative real numbers. Then the following two statements are equivalent.

(i) There existsC >0 such that for anyn∈N and any ε >0 P(max

`≤n|S`| ≥ε)≤Cε^−rX

`≤n

a_`.

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(ii) There existsC >0such that for any nondecreasing sequence(bn)n∈Nof positive real numbers, for anyn∈N and any ε >0

P

max`≤n|S`|b⁻¹_` ≥ε

≤Cε^−rX

`≤n

a`b^−r_` .

And next, they derived from it this SLLN.

Theorem 1.4. Let a_n and b_n are non-negative sequences of real numbers and let r >0. Suppose thatb_n is a positive non-decreasing, unbounded sequence of positive real numbers. Let us assume that

X

n

a_n b^r_n <∞

and there existsC >0 such that for anyn∈N and any ε >0 P

maxm≤n|Sm| ≥ε

≤C ε^−r X

m≤n

am.

Then

n→∞lim Sn

bn

= 0 a.s.

As said previously, this paper aims at generalizing the previous results in the following way. First, we give a random fields version for Tómács and Líbor [16] as a first generalization in Proposition 2.1. Next we show that our version of Hájek- Rényi type maximal inequality for probabilities for random fields is a generalization of that of Noszály and Tómács [14] and leads to a more general SLLN.

We apply our method for logarithmically weighted sums without any moment assumption and under general dependence conditions for random fields. This shows that the generalization is not trivial.

The paper is organized as follows. Section 2 is devoted to our main results, a Hájek-Rényi type maximal inequality for probabilities for random fields and automatically a strong law of large numbers are given. Section 3 includes their proofs. Section 4 including applications and illustration of our results, concludes the paper.

2. Results

We first give a Hájek-Rényi type maximal inequality for probabilities for random fields, as an extension of Proposition 1 in Noszály and Tómács [14] and of Theorem 2.1 in Tómács and Líbor [16].

Proposition 2.1. Letrbe a positive real number,a_nbe a nonnegatived-sequence.

Suppose that b_n is a positive, nondecreasingd-sequence of product type. Then the

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On general strong laws of large numbers for fields of random variables 7 following two statements are equivalent.

(i) There existsC >0 such that for anyn∈N^d and any ε >0 P(max

`≤n|S`| ≥ε)≤Cε^−rX

`≤n

a_`.

(ii) There exists C >0such for any n∈N^d and any ε >0 P

max`≤n|S`|b⁻¹_` ≥ε

≤4^dC ε^−r X

`≤n

a`b^−r_` .

We derive from this proposition a general strong law of large numbers for random fields which includes extensions of Theorem 3 in Noszály and Tómács [14] and of Theorem 2.4 in Tómács and Líbor [16]. But we need this lemma first.

Lemma 2.2 (Lemma 2 in Noszály and Tómács [14]). Let a_n be a nonnegative d- sequence and let b_n be a positive, nondecreasing, unbounded d-sequence of product type. Suppose thatP

n a_n

b^r_n <∞with a fixed realr >0. Then there exists a positive, nondecreasing, unbounded d-sequenceβ_n of product type for which

limn

βn

b_n = 0 and X

n

an

β^r_n <∞.

Here is our general strong law of large numbers.

Theorem 2.3. Let an be a non-negative d-sequence and let r > 0. Suppose that bnis a positive, non-decreasing, unbounded d-sequence of product type. If

X

n

an

b^r_n <∞

and there existsC >0 such that for anyn∈N^d and anyε >0 P

maxm≤n|Sm| ≥ε

≤C ε^−r X

m≤n

am

then

n→∞lim S_n bn

= 0 a.s.

3. Proofs of the main results

We will need Lemma 2.2 and these two following lemmas.

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Lemma 3.1. Let {Yk,k ∈N^d} be a field of random variables defined on a fixed probability space(Ω,F,P). Then for all x∈R,

P

sup

k

Y_k> x

= lim

n→∞P

max

k≤nY_k> x

. Proof. It is easy to see that, for allx∈R.

sup

k

Yk> x

=

∞

[

n=1

maxk≤nYk> x

.

Hence, by the monotone convergence theorem for probabilities, we get the state- ment.

Lemma 3.2. Let {Yk,k ∈N^d} be a field of random variables defined on a fixed probability space (Ω,F,P)and{εn,n∈N^d} a nondecreasing field of real numbers.

If

n→∞limP

sup

k

Yk> εn

= 0, then sup_kYk<∞ a.s.

Proof. By using the monotone convergence theorem for probabilities, we have P

∞

\

n=1

sup

k

Y_k> ε_n !

= lim

n→∞P

sup

k

Y_k> ε_n

= 0

which is equivalent toP(S∞

n=1(sup_k Yk≤εn)) = 1. This implies that there exists nω∈N^d for almost everyω∈Ωsuch thatsup_k Y_k(ω)≤ε_n_ω<∞.

We need more notation for the proofs. In N^d the maximum is defined coordinate-wise (actually we shall use it only for rectangles). Ifn= (n1, . . . , nd)∈N^d, then hni = Qd

i=1ni. A numerical sequence an,n ∈ N^d is called d-sequence. If an is a d-sequence then its difference sequence, i.e. the d-sequence bn for which P

m≤nb_m=a_n,n∈N^d, will be denoted by∆a_n(i.e.∆a_n=b_n). We shall say that a d-sequence an is of product type ifan =Qd

i=1a⁽ⁱ⁾ni, where a⁽ⁱ⁾ni (ni = 0,1,2, . . .) is a (single) sequence for each i = 1, . . . , d. Our consideration will be confined to normalizing constants of product type: bn will always denote bn =Qd

i=1b⁽ⁱ⁾ni, whereb⁽ⁱ⁾n_i (n_i= 0,1,2, . . .)is a nondecreasing sequence of positive numbers for each i= 1, . . . , d. In this case we shall say thatb_nis a positive nondecreasingd-sequence of product type. Moreover, if for eachi= 1, . . . , dthe sequenceb⁽ⁱ⁾n_i is unbounded, then b_n is called positive, nondecreasing, unboundedd-sequence of product type.

As usual, log⁺(x) := max{1,log(x)}, x >0and |logn|:=Qd

m=1log⁺n_m.

Proof of Proposition 2.1. It is clear that (ii) implies (i) by taking b_m_j = 1 for all m ∈ N^d and 1 ≤ j ≤ d. Now we turn to (i) =⇒ (ii). We can assume without

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On general strong laws of large numbers for fields of random variables 9 loss of generality that b0,j = 1 for 1 ≤ j ≤ d. If not, we would replace bm by Qd

j=1b_m,j/b_0,j,m∈N^d and (ii) would remain true with a new constant equal to Cb^−r₀ = C(Qd

j=1b0,j)^−r. Now consider a fixed n ∈ N^d and an arbitrary a real number c >1. Remark by the monotonicity of(bm)that bm_j ≥1 for allm∈N^d and that the sequence (c^p)p≥0 forms a partition of [1,+∞[. This implies that for any m ∈N^d, for any 1 ≤j ≤d, there exists a nonnegative integer ij such that cⁱ^j ≤b_m_j < cⁱ^j⁺¹. Thus fori= (i₁, . . . , i_d), we have that m∈ Ai={s∈N^d and cⁱ^j ≤b_s_j < cⁱ^j⁺¹, j= 1, . . . , d}. Since this holds for allm∈N^d, we get

N^d= [

i∈N^d

Ai. Let us restrict ourselves tom≤n, and let us define

Ai,n={s∈N^d,s≤n and cⁱ^j ≤bs_j < cⁱ^j⁺¹, j= 1, . . . , d}.

Sincec^p→ ∞asp→ ∞and form∈ Ai,n, for1≤j≤d,bs_j ≤bn_j ≤max{bn_k,1≤ k≤d}<∞, the setsAi,n are empty for large values ofi. Then putkn= max{i: Ai,n6=∅}<+∞and we have

[0, n] = [

i≤k_n

Ai,n.

It is also noticeable that if m≤s∈ Ai,n, then necessarily mis in someAi⁰,nwith i⁰ ≤i. As well let mi,n= maxAi,n≤nand defineDi,n=P

m∈Ai,nam where, by convention,Di,n= 0 andmi,n= (0, . . . ,0)whenAi,n=∅. From all that, we have

P

max

m≤n|S_m|b⁻¹_m ≥ε

≤ X

i≤kn

P

m∈Amax_i,n|S_m|b⁻¹_m ≥ε

.

Since form∈ Ai,n,bm=Qd

j=1bm_j ≥Qd

j=1cⁱ^j andAi,n⊂[0, mi,n], we get P

maxm≤n|S_m|b⁻¹_m ≥ε

≤ X

i≤kn

P

m∈Amax_i,n|S_m|b⁻¹_m ≥ε

≤

X

i≤kn

P



 max

m∈A_i,n|S_m| ≥ε

d

Y

j=1

cⁱ^j



. Now by applying (i) one arrives at

P

max

m≤n|Sm|b⁻¹_m ≥ε

≤Cε^−rX

i≤kn

d

Y

j=1

c^−ri^j X

m≤m_i,n

a_m≤

Cε^−rX

i≤kn

d

Y

j=1

c^−ri^j X

m≤i

D_m,n.

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By the remark made above,m≤m_i,n ∈ Ai,nimplies thatmis in someAs,nwhere s≤iand then by the definition of theDi,non hasP

m≤m_i,nam≤P

m≤iDm,nand next

P

max

m≤n|Sm|b⁻¹_m ≥ε

≤Cε^−rX

i≤kn

d

Y

j=1

c^−ri^j X

m≤i

D_m,

which becomes by a straightforward manipulations on the ranges of the sums, and wherek_n(j)stands for thej-th coordinate ofk_n,

P

maxm≤n|Sm|b⁻¹_m ≥ε

≤Cε^−r X

m≤kn

Dm,n

X

m≤i≤kn

d

Y

j=1

c^−ri^j ≤

Cε^−r X

m≤kn

Dm,n d

Y

j=1

X

m_j≤ij≤kn(j)

c^−ri^j =

Cε^−r X

m≤kn

Dm,n d

Y

j=1

c^−rm^j −c^−r(kⁿ^(j)+1)

1−c^−r ≤Cε^−r X

m≤kn

Dm,n d

Y

j=1

c^−rm^j 1−c^−r, sincec >1 andkn(j) + 1> mj. Now, at this last but one step, we have

P

max

m≤n|S_m|b⁻¹_m ≥ε

≤Cε^−r c^r

1−c^−r ^d

X

m≤kn

D_m,n

d

Y

j=1

c^−r(m^j⁺¹⁾≤

Cε^−r c^r

1−c^−r ^d

X

m≤kn

X

s∈Am,n

as d

Y

j=1

c^−r(m^j⁺¹⁾.

Finally, taking into account the fact that for s ∈ Am,n, c^m^j⁺¹ ≥bs_j, 1 ≤j ≤d, that is Qd

j=1c^r(m^j⁺¹⁾≥b^r_s, we arrive at P

maxm≤n|Sm|b⁻¹_m ≥ε

≤Cε^−r c^r

1−c^−r d

X

m≤kn

X

s∈Am,n

as

b^r_s ≤ Cε^−r

c^r 1−c^−r

^d X

m≤n

a_m b^r_m. Sincec is arbitraryc >1andminc>1 c^r

1−c^−r = 4, we achieve the proof by P

maxm≤n|S_m|b⁻¹_m ≥ε

≤4^d C ε^−r X

m≤n

am

b^r_m.

Proof of Theorem 2.3. Letβ_nbe the d-sequence obtained in the Lemma 2.2. Ac- cording to Proposition 2.1

P

max

`≤m|S`|β_`⁻¹≥ε_k

≤4^dCε^−r_k X

`≤m

a_`β^−r_` ∀m≤n.

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On general strong laws of large numbers for fields of random variables 11 By this fact we get for any fixedk∈N^d

P

sup

`≤m

|S`|β_`⁻¹≥εk

≤ lim

m→∞P

max`≤m|S`|β_`⁻¹≥εk

≤4^dCε^−r_k X

n

anβ^−r_n ,

where{εk,k∈N^d}a positive, nondecreasing, unbounded field of real numbers. So we have by Lemma 2.2

lim

k→∞P

sup

`

|S`|β_`⁻¹≥ε_k

= 0.

Using Lemma 3.1 P

sup

`

|S_`|β⁻¹_` ≥ε_k for allk∈N^d

= 0.

So we have by Lemma 3.2 sup_`|S`|β_`⁻¹<∞a.s. Finally by Lemma 2.2 0≤ |Sn|

b_n = |Sn| β_n

βn

b_n ≤sup

`

|S_`|β_`⁻¹βn

b_n →0 a.s.

4. Conclusion

4.1. A first application: Logarithmically weighted sums

The following result is an extension of Theorem 7 in Noszály and Tómács [14] and of Theorem 4.2 in Fazekas et al. [5]. In this Theorem, we do not need any moment assumption in contrary of these above cited theorems.

Theorem 4.1. Let {X_n,n∈N^d} be a field of random variables. Letr >1. We assume there exists C >0 such that for anym∈N^d and anyε >0

P



max`≤m

X

k≤`

Xk

hki ≥ε



≤Cε^−rX

`≤m

1 h`i. Then

1

|logn|

X

k≤n

Xk

hki →0 (n→ ∞) a.s.

Proof. Let us apply Theorem 2.3 with an = _hni¹ and bn = |logn|. The proof is achieved by remarking that for r >1

X

n

an

b^r_n =X

n

1

|logn|^r 1 hni<∞.

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4.2. A second application

By using Markov’s Inequality and applying our results (see Theorem 2.3), under the same assumptions in Noszály and Tómács [14], we rediscover their results.

Acknowledgement. The paper was finalized while the second author was vis- iting MAPMO, University of Orléans, France, in 2011. He expresses her warm thanks to responsibles of MAPMO for kind hospitality. The authors also thank the referee for his valuable comments and suggestions.

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Annales Mathematicae et Informaticae 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

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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 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 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 ∂}_∂x_i 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.

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Harmonic sections on the tangent bundle of order two 17 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 ∂}_∂x_i ∈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^h^p (2.3) γ(G) =Gh₁h₂...h_py^h¹dx^h²⊗ · · · ⊗dx^h^p (2.4) whereF =F_i^j

1...i_p

∂

∂x^j ⊗dxⁱ¹⊗ · · · ⊗dxⁱ^p andG=Gi₁...i_pdxⁱ¹⊗ · · · ⊗dxⁱ^p. 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

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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

j₀¹f =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)

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Harmonic sections on the tangent bundle of order two 19

=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).

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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

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Harmonic sections on the tangent bundle of order two 21 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

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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,X_x₎+Yⁱ(x)∂X^k

∂xⁱ (x) ∂

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

(d_xX(Y_x))^h=Yⁱ(x) ∂

∂xⁱ|_(x,X_x₎−Yⁱ(x)X^j(x)Γ^k_ij(x) ∂

∂y^k|_(x,X_x₎

=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,X_x₎

= (∇YX)^V_(x,X_x₎.

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).

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Harmonic sections on the tangent bundle of order two 23 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_e_iσ)^Vi

σ(x)

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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+∇_(∇_ei_X_σ₎1e⁰_i +∇_(∇_ei_Y_σ₎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.