ON A CLASS OF ALMOST CONTACT STRUCTURES ON T

on the occasion of his 85^thbirthday

ON A CLASS OF ALMOST CONTACT STRUCTURES ON T

M

ADRIAN SANDOVICI

A new class of almost contact structures on second order tangent bundle built on a Riemannian space is defined and studied in this paper. In particular, characteriza- tions for the integrability and normality of this class of almost contact structures are given. Also, compatible linear connections with this class of almost contact structures are introduced and certain characterization of them is obtained.

AMS 2010 Subject Classification: 53B40, 53C60, 58B20.

Key words:second order tangent bundle; almost contact structure; linear connec- tion.

1. INTRODUCTION

The generalized Lagrange geometry of second order was defined and stu- died by R. Miron [6, 7] and represents the geometry of generalized Lagrangians modeled on the second order tangent bundle T²M, p, M

. These spaces are useful in the study of the geometry of higher-order Lagrangians [6, 7], for the prolongation of Riemannian, Finslerian and Lagrangian structures [6, 7], for the study of stationary curves [9], and for the development of a gauge theory having the second order tangent bundle as the geometrical model [3, 10, 12].

The term “homogeneity” has been discussed in Miron’s papers [4, 5] where new geometrical models on Riemannian spaces and on Finslerian spaces are also introduced, respectively. In [12, 14] an extension of Miron’s theory of homogeneity to the second order tangent bundle is presented.

The main goal of this note is to define and study a new class of almost contact structures on second order tangent bundle built on a Riemannian space. In particular, characterizations for the integrability and normality of this class of almost contact structures are given. Also, compatible linear con- nections with this class of almost contact structures are introduced and certain characterization of them is obtained.

REV. ROUMAINE MATH. PURES APPL.,57(2012),2, 159-164

This paper can be seen as a natural continuation of our previous works [11, 14]. It should be also mentioned that the basic concepts and the notations coincide with those from [11, 14].

2. PRELIMINARIES

ConsiderRⁿ= (M, γ) a Riemannian space generated by a real, differen- tiable,n-dimensional manifoldM and by a Riemannian metricγ onM, given by the local components (γij(x)), x ∈ U ⊂ M. It is possible to extend γ to p⁻¹(U)⊂E =T²M by

(2.1) (γ_ij◦p) (u) =γ_ij(x), u∈p⁻¹(U), p(u) =x.

In this case γ_ij ◦p are the local components of a tensor field onE. Usually, we write these local components with γij as well. Furthermore, with γ_ij^k(x) we will denote the Christoffel symbols of the second species of the metric γ and with r_ijh^k (x) we will denote the local components of the curvature tensor field of the metric γ. It is possible to introduce on E a nonlinear connection determined only by this metric, cf. [6]. Moreover, the coefficients of connection are determined by the following relations (see also [11])

N_(1)j⁽⁰⁾ⁱ x, y⁽¹⁾

=γ_j0ⁱ , (2.2)

N_(2)j⁽⁰⁾ⁱ x, y⁽¹⁾, y⁽²⁾

= 1 2

∂γ_j0ⁱ

∂x^py^(1)p+γ_0mⁱ ·γ^m_j0

! +γ_jⁱ_¯0, (2.3)

where the notation “ 0 ” stands for the contraction by y⁽¹⁾

and the notation

“ ¯0 ” stands for the contraction by y⁽²⁾ .

The nonlinear connectionN assures the existence of a basis δk, δ_k⁽¹⁾, δ_k⁽²⁾ adapted to the tangent space T_uE. The vector fields of the adapted basis are defined with the help of the following relations

δ_k= ∂

∂x^k −N_(1)kⁱ ∂

∂y⁽¹⁾ⁱ −N_(2)kⁱ ∂

∂y⁽²⁾ⁱ, (2.4)

δ⁽¹⁾_k = ∂

∂y⁽¹⁾ⁱ −N_(1)kⁱ ∂

∂y⁽²⁾ⁱ, δ_k⁽²⁾ = ∂

∂y^(2)k. (2.5)

For further developments, we need the following result.

Theorem 2.1. The Lie brackets of the vector fields of the adapted basis δ_k, δ_k⁽¹⁾, δ⁽²⁾_k

are given by

(2.6)

δj, δk

=Rⁱ_(01)jk·δ⁽¹⁾_i +Rⁱ_(02)jk·δ_i⁽²⁾,

(2.7)

δ_j, δ⁽¹⁾_k

=Bⁱ_(11)jk·δ⁽¹⁾_i +B_(12)jkⁱ ·δ_i⁽²⁾,

(2.8)

δj, δ⁽²⁾_k

=Bⁱ_(21)jk·δ⁽¹⁾_i +B_(22)jkⁱ ·δ_i⁽²⁾,

(2.9)

δ_j⁽¹⁾, δ_k⁽¹⁾

=Rⁱ_(12)jk ·δ_i⁽²⁾,

δ⁽¹⁾_j , δ⁽²⁾_k

=B_(21)jkⁱ ·δ⁽²⁾_i , where

(2.10) Rⁱ_(01)jk =r_0jkⁱ , Rⁱ_(12)jk = 0, (2.11) Rⁱ_(02)jk = 1

2·∂rⁱ_0jk

∂x^p ·y^(1)p+1

2· γ_0mⁱ ·r_0jk^m +γ_0j^m·rⁱ_0km+γ_0k^m·rⁱ_0mj +rⁱ¯0jk, (2.12) B_(11)jkⁱ =B_(22)jkⁱ =γ_jkⁱ , B_(21)jkⁱ = 0,

(2.13) B_(12)jkⁱ = 1

2·rⁱ_0jk+γ_0mⁱ ·γ_jk^m.

3. β ALMOST CONTACT STRUCTURES ON TTT²²²MMM

Assume that β :R→ (0,+∞) is a smooth function. Define locally the following F(E) linear map F^β :X(E)→X(E) by

(3.1) F^β(δ_i) =− 1

β(F²)·δ⁽²⁾_i , F^β(δ_i⁽¹⁾) = 0, F^β(δ⁽²⁾_i ) =β(F²)·δ_i, where F²=γ_ij ·y⁽¹⁾ⁱy^(1)j.

The linear structure defined by (3.1) is called theβ almost contact struc- ture onT²M. Some basic properties of this structure are listed within the next result.

Theorem 3.1. The linear map F^β has the following properties:

1. F^β is globally defined on E;e 2. F^β is a (1,1)tensor field on E;e

3. F^β is an almost contract structure on E;e 4. (F^β)³+F^β = 0;

5. F^β depends only on the Riemannian metricγ and on a strictly positive smooth function β;

6. ker(F^β) =N₁, Im(F^β) =N₀+V₂; 7. rank(F^β) = 2n.

It is easily seen thatF^β can be represented as

(3.2) F^β =− 1

β(F²) ·δ⁽²⁾_i ⊗dxⁱ+β(F²)·δ_i⊗δy⁽²⁾ⁱ.

Definition 3.1. The almost contact structure F^β is said to be normal if the following relation holds true

(3.3) Ne_Fβ(X, Y) =N_Fβ(X, Y) + 2·

n

X

i=1

d(δy⁽¹⁾ⁱ)(X, Y)·δ⁽¹⁾_i = 0, for allX,Y ∈X(E), wheree N_F^β stands for the Nijenhuis tensor field associated toF^β.

One obtains the following characterization for the normality of the β almost contact structure F^β.

Theorem3.2. The almost contact structureF^β is normal if and only if β is a positive constant and γ_jkⁱ (x) = 0.

The next result is an immediate consequence of a classical result due to T. Van Duc.

Theorem3.3. The almost contact structureF^β is integrable if and only if N_(Fβ)² = 0.

Using Theorem 3.3, a direct computation using local coordinates leads to the following characterization for the integrability of the β almost contact structure.

Theorem3.4. The almost contact structureF^β is integrable if and only if Rⁱ_(01)kh =rⁱ_jhk= 0.

In other words, Theorem 3.4 says that the almost contact structure F^β is integrable if and only if the underlying Riemannian space (M, γ) is flat.

4. COMPATIBLE CONNECTIONS WITH β CONTACT STRUCTURES With respect to the adapted basis δ_k, δ_k⁽¹⁾, δ⁽²⁾_k

, any linear connection D on E can be represented as follows

(4.1) D_δkδ_j =L^(H)i_jk ·δ_i+L⁽¹⁾ⁱ_jk ·δ_i⁽¹⁾+L⁽²⁾ⁱ_jk ·δ_i⁽²⁾, (4.2) Dδ_kδ_j⁽¹⁾ =L⁽³⁾ⁱ_jk ·δi+L^(v_jk¹⁾ⁱ·δ_i⁽¹⁾+L⁽⁴⁾ⁱ_jk ·δ_i⁽²⁾, (4.3) D_δkδ_j⁽²⁾ =L⁽⁵⁾ⁱ_jk ·δ_i+L⁽⁶⁾ⁱ_jk ·δ_i⁽¹⁾+L^(v_jk²⁾ⁱ·δ_i⁽²⁾, (4.4) D_δ(1)

k

δj =F_jk^(H)i·δi+F_jk⁽¹⁾ⁱ·δ_i⁽¹⁾+F_jk⁽²⁾ⁱ·δ⁽²⁾_i ,

(4.5) D

δ⁽¹⁾_k δ_j⁽¹⁾=F_jk⁽³⁾ⁱ·δ_i+F_jk^(v1)i·δ_i⁽¹⁾+F_jk⁽⁴⁾ⁱ·δ⁽²⁾_i , (4.6) D_δ(1)

k

δ_j⁽²⁾=F_jk⁽⁵⁾ⁱ·δi+F_jk⁽⁶⁾ⁱ·δ_i⁽¹⁾+F_jk^(v2)i·δ⁽²⁾_i , (4.7) D_δ(2)

k

δj =C_jk^(H)i·δi+C_jk⁽¹⁾ⁱ·δ_i⁽¹⁾+C_jk⁽²⁾ⁱ·δ_i⁽²⁾,

(4.8) D

δ⁽²⁾_k δ_j⁽¹⁾=C_jk⁽³⁾ⁱ·δ_i+C_jk^(v1)i·δ_i⁽¹⁾+C_jk⁽⁴⁾ⁱ·δ_i⁽²⁾,

(4.9) D

δ⁽²⁾_k δ_j⁽²⁾=C_jk⁽⁵⁾ⁱ·δ_i+C_jk⁽⁶⁾ⁱ·δ_i⁽¹⁾+C_jk^(v2)i·δ_i⁽²⁾.

The set consisting of the functionsL^(H_jk⁾ⁱ, . . . , C_jk^(v2)i represents the set of the coefficients of the linear connection D.

Definition 4.1. A linear connection D on E =T²M is said to be com- patible with theβ contact structure F^β ifD_XF^β = 0, for allX ∈X(E).

Concerning the notion of compatible connection with the β almost con- tact structure F^β, the following result can be proved.

Theorem4.1. A linear connectionDonE is compatible with aβ almost contact structure F^β if and only if the coefficients of the linear connection satisfy the following relations

L^(H_jk⁾ⁱ=L^(V_jk²⁾ⁱ, L⁽⁵⁾ⁱ_jk =−β²·L⁽²⁾ⁱ_jk , L⁽¹⁾ⁱ_jk =L⁽³⁾ⁱ_jk =L⁽⁴⁾ⁱ_jk =L⁽⁶⁾ⁱ_jk = 0, F_jk^(V2)i =F_jk^(H)i+ 2· β

β⁰ ·y_k⁽¹⁾·δ_jⁱ,

F_jk⁽⁵⁾ⁱ=−β²·F_jk⁽²⁾ⁱ, F_jk⁽¹⁾ⁱ=F_jk⁽³⁾ⁱ =F_jk⁽⁴⁾ⁱ =F_jk⁽⁶⁾ⁱ = 0,

C_jk^(V2)i=C_jk^(H)i, C_jk⁽⁵⁾ⁱ=−β²·C_jk⁽²⁾ⁱ, C_jk⁽¹⁾ⁱ=C_jk⁽³⁾ⁱ=C_jk⁽⁴⁾ⁱ=C_jk⁽⁶⁾ⁱ= 0.

The next result provides necessary and sufficient conditions for ad-linear connection to be compatible with an almost contact structure F^β.

Corollary 4.1. A d-linear connection D onE is compatible with a β almost contact structure F^β if and only if the coefficients of the linear connec- tion satisfy the following relations

L^(H)i_jk =L^(V_jk²⁾ⁱ, F_jk^(H)i =F_jk^(V2)i−2· β

β⁰ ·y⁽¹⁾_k ·δ_jⁱ, C_jk^(H)i=C_jk^(V2)i.

Acknowledgements. This work was supported by AM-POSDRU, project number:

POSDRU/89/1.5/S/49944.

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Received 3 March 2012 “Al.I. Cuza” University

Department of Sciences Str. Lasc˘ar Catargi 54

700107 Ia¸si, Romania adrian.sandovici@uaic.ro