Lie derivatives of almost contact structure and almost paracontact structure with respect to...

http://dx.doi.org/10.20852/ntmsci.2016115657

Lie derivatives of almost contact structure and almost

paracontact structure with respect to

X

C

and

X

V

on

tangent bundle

T

(

M

)

Hasim Cayir and Gokhan Koseoglu

Department of Mathematics, Giresun University, Giresun, Turkey

Received: 13 November 2015, Revised: 10 December 2015, Accepted: 4 January 2016 Published online: 31 January 2016.

Abstract: The differential geometry of tangent bundles was studied by several authors, for example: D. E. Blair [1], V. Oproiu [3], A. Salimov [5], Yano and Ishihara [8] and among others. It is well known that differant structures deffined on a manifoldMcan be lifted to the same type of structures on its tangent bundle. Our goal is to study Lie derivatives of almost contact structure and almost paracontact structure with respect toXCandXVon tangent bundleT(M). In addition, this Lie derivatives which obtained shall be studied for some special values.

Keywords: Lie derivative, almost contact structure, almost paracontact structure, tangent bundle.

1 Introduction

LetMbe ann−dimensional differentiable manifold of classC∞and letTp(M)be the tangent space ofMat a pointpof

M.Then the set [8]

T(M) = ∪

p∈MTp(M) (1)

is called the tangent bundle over the manifoldM.For any point ˜pof T(M),the correspondence ˜p→p determines the bundle projectionπ:T(M)→M,thusπ(p˜) =p,whereπ:T(M)→M defines the bundle projection ofT(M)overM. The setπ−1_{(p)is called the fibre overp∈MandMthe base space.}

Suppose that the base spaceMis covered by a system of coordinate neighbour-hoods{U;xh}

,where(x

h_{)is a system of}

local coordinates defined in the neighbour-hood U of M. The open set π−1_{(U)⊂T(M) is naturally differentiably} homeomorphic to the direct productU×Rn,R

n_{being then−dimensional vector space over the real field R,in such a}

way that a point ˜p∈Tp(M)(p∈U)is represented by an ordered pair(P,X)of the point p∈U,and a vectorX ∈R

n

,whose components are given by the cartesian coordinates(yh)of ˜pin the tangent spaceTp(M)with respect to the natural

base{∂h},where∂h= ∂

∂xh. Denoting by(x

h_{)the coordinates of p=π(p˜)inU and establishing the correspondence}

(xh

,y

h_)→p˜∈π−1_(U)

,we can introduce a system of local coordinates(x

h_,yh_{)in the open setπ}−1_(U)⊂T(M)

.Here we cal(xh,y

h_{)the coordinates inπ}−1_{(U)induced from(x}h_{)or simply, the induced coordinates inπ}−1_(U)

.

We denote by ℑr

s(M) the set of all tensor fields of classC∞and of type(r,s)inM.We now putℑ(M) = ∞ ∑

r,s=0

ℑr s(M), which is the set of all tensor fields inM.Similarly, we denote byℑr

s(T(M))andℑ(T(M))respectively the corresponding

2 Vertical lifts

If f is a function inM,we write fvfor the function inT(M)obtained by forming the composition ofπ:T(M)→Mand f:M→R,so that

fv=f oπ. (2)

Thus, if a point ˜p∈π−1_{(U)has induced coordinates(x}h

,y

h₎

,then

fv(p˜) =fv(x,y) =f oπ(p˜) =f(p) =f(x). (3)

Thus, the value offv(p˜)is constant along each fibreTp(M)and equal to the value f(p). We call fvthe vertical lift of the

functionf [8].

Let ˜X∈ℑ1

0(T(M))be such that ˜X fv=0 for all f ∈ℑ00(M).Then we say that ˜X is a vertical vector field. Let

[

˜ Xh

˜ Xh¯

]

be

components of ˜X with respect to the induced coordinates. Then ˜X is vertical if and only if its components inπ−1_(U)

satisfy _[

˜ Xh

˜ Xh¯

]

=

[

0 Xh¯

]

. (4)

Suppose thatX∈ℑ1

0(M),so that is a vector field inM.We define a vector fieldX

v_inT(M)by

Xv(ι ω) = (ωX)v ₍₅₎

ωbeing an arbitrary 1−form inM.We calXvthe vertical lift ofX[8].

Let ˜ω ∈ℑ0

1(T(M))be such that ˜ω(X)v=0 for allX ∈ℑ10(M).Then we say that ˜ω is a vertical 1−form inT(M).We define the vertical liftωv_{of the 1−formω by}

ωv_{= (ω}

i)v(dxi)v (6)

in each open setπ−1_(U)

,where(U;x

h_{)is coordinate neighbourhood inMandωis given byω=ω}

idxiinU. The vertical

liftωv_{ofωwith lokal expressionω=ω}

idxi has components of the form

ωv_:(ωi

,0) (7)

with respect to the induced coordinates inT(M).

Vertical lifts to a unique algebraic isomorphism of the tensor algebraℑ(M)into the tensor algebraℑ(T(M))with respect to constant coefficients by the conditions

(P⊗Q)V _=PV_⊗QV

, (P+R)

V_=PV_+RV ₍₈₎

P,QandRbeing arbitrary elements ofℑ(M). The vertical liftsF

V _{of an elementF∈ℑ}1

1(M)with lokal componentsFih

has components of the form [8]

FV:

(

0 0 F_ih0

)

Vertical lift has the following formulas ([4],[8]):

(f X)v=fvXv,I

v_Xv₌

0,η

v_(Xv_{) =}

0 (10)

(fη)v=fvηv

,[X

v_,Yv_{] =0}

,ϕ

v_Xv₌₀

Xvfv=0,X

v_fv₌₀

hold good, where f∈ℑ0

0(Mn),X,Y∈ℑ

1

0(Mn),η∈ℑ

0

1(Mn),ϕ∈ℑ

1

1(Mn),I=idM_n.

3 Complete lifts

Iff is a function inM,we write f

c_{for the function inT(M)defined by}

fc=ι(d f)

and callfcthe comple lift of the function f.The complete lift f

c_{of a function f has the lokal expression}

fc=yi∂if =∂f (11)

with respect to the induced coordinates inT(M), where∂f denotesyi∂if.

Suppose thatX∈ℑ1

0(M). then we define a vector fieldXcinT(M)by

Xcfc= (X f)c

, (12)

f being an arbitrary function inM and callXc_{the complete lift ofX inT(M)([2],[8])}

.The complete liftX

c_{ofX with}

componentsxhinMhas components

Xc=

(

Xh ∂Xh

)

(13)

with respect to the induced coordinates inT(M).

Suppose thatω∈ℑ0

1(M),then a 1−formω

c_{inT(M)defined by}

ωc_(Xc_{) = (ωX)}c

. (14)

X being an arbitrary vector field inM.We callωc_{the complete lift ofω.The complete liftω}c_{ofω with componentsω} i

inMhas components of the form

ωc_{:(∂ ω}

i,ωi) (15)

with respect to the induced coordinates inT(M)[2].

The complete lifts to a unique algebra isomorphism of the tensor algebraℑ(M)into the tensor algebraℑ(T(M))with respect to constant coefficients, is given by the conditions

(P⊗Q)C=PC⊗QV+PV⊗QC, (P+R)

C_=PC_+RC

whereP,QandRbeing arbitrary elements ofℑ(M). The complete liftsF

C_{of an elementF∈ℑ}1

1(M)with lokal components

Fh

i has components of the form

FC:

(

F_ih 0 ∂F_ihF_ih

)

. (17)

In addition, we know that the complete lifts are defined by ([4],[8]):

(f X)c=fcXv+fvXc= (X f)c

, (18)

Xcfv= (X f)v,η

v_(xc_{) = (η(x))}v

,

Xvfc= (X f)v,ϕ

v_Xc_{= (ϕX)}v

, ϕcXv= (ϕX)v,(ϕX)

c_=ϕc_Xc

, ηv_(Xc_{) = (η(X))}c

,η

c_(Xv_{) = (η(X))}v

,

[Xv,Yc] = [X,Y]v,I

c_=I,Iv_Xc_=Xv

,[X

c_,Yc_{] = [X,Y]}c

.

Let M be ann−dimensional diferentiable manifold. Differantial transformationD=L_{X is called Lie derivation with}

respect to vector fieldX∈ℑ1 0(M)if

L_Xf =X f,∀f ∈ℑ

0

0(M), (19)

L_{XY = [X,Y]},∀X,Y ∈ℑ

1 0(M).

[X,Y]is called by Lie bracked. The Lie derivativeL_{XFof a tensor fieldFof type(1}

,1)with respect to a vector fieldXis defined by ([8])

(L_{XF)Y= [X},FY]−F[X,Y].

Proposition 1.For any X∈ℑ1

0(Mn), f ∈ℑ00(Mn)andLX is the Lie derivation with respect to vector field X [8]

(i) L_Xvfv=0, (ii) L_Xvfc= (L_Xf)v, (iii) L_Xcfv= (L_Xf)v, (iv) L_Xcfc= (L_Xf)c.

Proposition 2.For any X,Y ∈ℑ1

0(Mn)andLXis the Lie derivation with respect to vector field X [8]

(i) L_XvYv=0, (ii) L_XvYc= (∇_XY)v, (iii) L_XcYv= (∇_XY)v, (iv) L_XcYc= (∇_XY)c.

4 Main results

Definition 1. Let an n−dimensional differentiable manifold M be endowed with a tensor fieldϕof type(1,1),a vector fieldξ, a1−formη, I the identity and let them satisfy

ϕ2_{=−I+η⊗ξ}

Then(ϕ,ξ,η)define almost contact structure onM([4],[7],[8]). From (20), we get on taking complete and vertical lifts

(ϕc₎2_=−I+ηv_⊗ξc_+ηc_⊗ξv

, (21)

ϕc_ξv₌₀

,ϕ

c_ξc₌₀

,η

v_oϕc₌₀

, ηc_oϕc₌₀

,η

v_(ξv_{) =0}

,η

v_(ξc_{) =1}

, ηc_(ξv_{) =}

1,η

c_(ξc_{) =}

0.

We now define a(1,1)tensor fieldJonℑ(M)by

J=ϕc_−ξv_⊗ηv_+ξc_⊗ηc

. (22)

Then it is easy to show thatJ2Xv=−XvandJ2Xc=−Xc,which give thatJis an almost contact structure onℑ(M).We get from (22)

JXv= (ϕX)v+ (η(X))vξc

,

JXc= (ϕX)c−(η(X))vξv+ (η(X))cξc

for anyX∈ℑ1 0(M)[4].

Theorem 1. ForL_{Xthe operator Lie derivation with respect to X , J∈ℑ}1

1(ℑ(M))defined by (22) andη(Y) =0,we have (i) (L_XvJ)Yv=0,

(ii) (L_XvJ)Yc= ((L_Xϕ)Y)v+ ((L_Xη)Y)vξc, (iii) (L_XcJ)Yv= ((L_Xϕ)Y)v+ ((L_Xη)Y)vξc, (iv) (Lc

XcJ)Yc= ((LXϕ)Y)c−((LXη)Y)vξv+ ((LXη)Y)cξc, whereX,Y ∈ℑ1

0(M),a tensor fieldϕ∈ℑ

1

1(M),a vector fieldξ and a 1−formη∈ℑ

0 1(M).

Proof.ForJ=ϕc_−ξv_⊗ηv_+ξc_⊗ηc_{andη(Y) =0, we get}

(i) (L_XvJ)Yv=L_Xv(ϕc−ξv⊗ηv+ξc⊗ηc)Yv−(ϕc−ξv⊗ηv+ξc⊗ηc)L_XvYv

=L_Xv(ϕY)v−L_Xv(ηv(Y)v)ξv+L_Xv(η(Y))vξc

=0,

(ii) (L_XvJ)Yc=L_Xv(ϕc−ξv⊗ηv+ξc⊗ηc)Yc−(ϕc−ξv⊗ηv+ξc⊗ηc)L_XvYc

=L_XvϕcYc−L_Xv(ηY)vξv+L_Xv(η(Y))cξc−ϕcL_XvYc

+ηv_(L

XY)vξv−(η(LXY))vξc

= (L_Xvϕc)Yc+ϕc(L_XvYc)−ϕcL_XvYc−(L_X(η(Y)))vξc+ ((L_Xη)Y)vξc

= (L_Xϕ)Y)v_{+ ((L}

Xη)Y)vξc,

(iii) (L_XcJ)Yv=L_Xc(ϕc−ξv⊗ηv+ξc⊗ηc)Yv−(ϕc−ξv⊗ηv+ξc⊗ηc)L_XcYv

=L_XcϕcYv−L_Xc(ηv(Y)v)ξv+L_Xc(η(Y))vξc−ϕcL_XcYv

+ηv_(L

XY)vξv−(η(LXY))vξc

= (L_Xcϕc)Yv+ϕc(L_XcYv)−ϕcL_XcYv−(L_X(η(Y)))vξc+ (L_Xη)Y)vξc

= (L_Xϕ)Y)v_{+ ((L}

Xη)Y)vξc,

(iv) (L_XcJ)Yc=L_Xc(ϕc−ξv⊗ηv+ξc⊗ηc)Yc−(ϕc−ξv⊗ηv+ξc⊗ηc)L_XcYc

=L_XcϕcYc−L_Xc((ηY)v)ξv+L_Xc(η(Y))cξc−ϕcL_XcYc

+ (η(L_XY))v_ξv_−(η(L XY))cξc

−(L_X(η(Y)))c_ξc_{+ ((L}

Xη)Y)cξc

= (L_Xϕ)Y)c_−((L

Xη)Y)vξv+ ((LXη)Y)cξc.

Corollary 1.If we put Y=ξ, i.e.η(ξ) =1andξ has the conditions of (20),then we get different results

(i) (L_XvJ)ξv= (L_Xξ)v,

(ii) (L_XvJ)ξc= ((L_Xϕ)ξ)v+ (((L_Xη))ξ)vξc,

(iii) (L_XcJ)ξv= ((L_Xϕ)ξ)v+ (L_Xξ)c+ ((L_Xη)ξ)vξc,

(iv) (L_XcJ)ξc= (L_Xϕ)ξ)c−(L_Xξ)v−((L_Xη)ξ)vξv+ ((L_Xη)ξ)cξc.

Definition 2. Let an n−dimensional differentiable manifold M be endowed with a tensor fieldϕ of type(1,1), a vector fieldξ, a1−formη, I the identity and let them satisfy

ϕ2_=I−η⊗ξ

, ϕ(ξ) =0, ηoϕ=0, η(ξ) =1. (23)

Then(ϕ,ξ,η)define almost paracontact structure onM([4],[7]). From (23), we get on taking complete and vertical lifts

(ϕc₎2_=I−ηv_⊗ξc_−ηc_⊗ξv

, (24)

ϕc_ξv₌₀

,ϕ

c_ξc₌₀

,η

v_oϕc₌₀

, ηc_oϕc₌₀

,η

v_(ξv_{) =0}

,η

v_(ξc_{) =1}

, ηc_(ξv_{) =}

1,ηc_(ξc_{) =}

0.

We now define a(1,1)tensor fieldJeonℑ(M)by

e

J=ϕc_−ξv_⊗ηv_−ξc_⊗ηc

. (25)

Then it is easy to show thatJe2_Xv_=Xv_andJe2_Xc_=Xc

,which give thatJeis an almost product structure onℑ(M).We get from (25) for anyX∈ℑ1

0(M).

e

JXv= (ϕX)v−(η(X))vξc

,

e

JXc= (ϕX)v−(η(X))vξv_−(η(X))c_ξc

.

Theorem 2. ForL_Xthe operator Lie derivation with respect to X ,Je∈ℑ1

1(ℑ(M))defined by (25)andη(Y) =0, we have (i) (L_XvJe)Yv=0,

(ii) (L_XvJe)Yc= ((L_Xϕ)Y)v−((L_Xη)Y)vξc, (iii) (L_XcJe)Yv= ((L_Xϕ)Y)v−((L_Xη)Y)vξc,

(iv) (L_XcJe)Yc= ((L_Xϕ)Y)c−((L_Xη)Y)vξv−((L_Xη)Y)cξc,

whereX,Y ∈ℑ1

0(M),a tensor fieldϕ∈ℑ

1

1(M),a vector fieldξ∈ℑ

1

0(M)and a 1−formη∈ℑ01(M).

Proof.ForJe=ϕc_−ξv_⊗ηv_−ξc_⊗ηc_{andη(Y) =0}

,we get

(i) (L_XvJe)Yv=L_Xv(ϕc−ξv⊗ηv−ξc⊗ηc)Yv−(ϕc−ξv⊗ηv−ξc⊗ηc)L_XvYv

=L_Xv(ϕY)v−L_Xv(ηv(Y)v)ξv−L_Xv(η(Y))vξc

=0,

(ii) (L_XvJe)Yc=L_Xv(ϕc−ξv⊗ηv−ξc⊗ηc)Yc−(ϕc−ξv⊗ηv−ξc⊗ηc)L_XvYc

+ηv_(L

XY)vξv+ (η(LXY))vξc

= (L_Xvϕc)Yc+ϕc(L_XvYc)−ϕc(L_XY)v+ (L_X(η(Y)))vξc−((L_Xη)Y)vξc

= (L_Xϕ)Y)v_−((L

Xη)Y)vξc,

(iii) (L_XcJe)Yv=L_Xc(ϕc−ξv⊗ηv−ξc⊗ηc)Yv−(ϕc−ξv⊗ηv−ξc⊗ηc)L_XcYv

=L_XcϕcYv−L_Xc(ηv(Y)v)ξv−L_Xc(η(Y))vξc−ϕcL_XcYv

+ηv_(L

XY)vξv+ (η(LXY))vξc

= (L_Xcϕc)Yv+ϕc(L_XcYv)−ϕcL_XcYv+ (L_X(η(Y)))vξc−(L_Xη)Y)vξc

= (L_Xϕ)Y)v_−((L

Xη)Y)vξc,

(iv) (L_XcJe)Yc=L_Xc(ϕc−ξv⊗ηv−ξc⊗ηc)Yc−(ϕc−ξv⊗ηv−ξc⊗ηc)L_XcYc

=L_XcϕcYc−L_Xc((ηY)v)ξv−L_Xc(η(Y))cξc−ϕcL_XcYc

+ (η(L_XY))v_ξv_{+ (η(L} XY))cξc

= (L_Xcϕc)Yc+ϕc(L_XcYc)−ϕcL_XcYc+ (L_X(η(Y)))vξv−((L_Xη)Y)vξv

+ (L_X(η(Y)))c_ξc_−((L

Xη)Y)cξc

= (L_Xϕ)Y)c_−((L

Xη)Y)vξv−((LXη)Y)cξc.

Corollary 2.If we put Y=ξ, i.e.η(ξ) =1andξ has the conditions of (23),then we have

(i) (L_XvJe)ξv=−(L_Xξ)v,

(ii) (L_XvJe)ξc= ((L_Xϕ)ξ)v−(((L_Xη))ξ)vξc,

(iii) (L_XcJe)ξv= ((L_Xϕ)ξ)v−(L_Xξ)c−((L_Xη)ξ)vξc,

(iv) (L_XcJe)ξc= (L_Xϕ)ξ)c−(L_Xξ)v−((L_Xη)ξ)vξv−((L_Xη)ξ)cξc.

References

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(50) (1984) 93 – 97.

[5] A.A.Salimov, Tensor Operators and Their applications, Nova Science Publ., New York (2013).

[6] S.Sasaki, On The Differantial Geometry of Tangent Boundles of Riemannian Manifolds, Tohoku Math. J., no.10(1958) 338-358. [7] A.A.Salimov, H.C¸ ayır, Some Notes On Almost Paracontact Structures, Comptes Rendus de 1’Acedemie Bulgare Des Sciences,

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