On Tachibana and Vishnevskii Operators Associated with Certain Structures in the Tangent Bundle

(1)

ISSN Online: 2327-4379 ISSN Print: 2327-4352

DOI: 10.4236/jamp.2018.610168 Oct. 10, 2018 1968 Journal of Applied Mathematics and Physics

On Tachibana and Vishnevskii Operators

Associated with Certain Structures in the

Tangent Bundle

Lovejoy S. Das

1

, Mohammad Nazrul Islam Khan

2

1_{Department of Mathematics, Kent State University, Tuscarawas, New Philadelphia, OH, USA} 2_{Department of Computer Engineering, College of Computer, Qassim University, Buraidah, KSA}

Abstract

The aim of the present work is to study the complete, vertical and horizontal lifts using Tachibana and Visknnevskii operators along generalized almost r-contact structure in tangent bundle. We also prove certain theorems on Tachibana and Visknnevskii operators with Lie derivative and lifts.

Keywords

Tangent Bundle, Vertical Lift, Complete Lift, Lie Derivative, Tachibana Operator, Vishnevskii Operator

1. Introduction

Let M be an n-dimensional differentiable manifold and let T M

( )

=



p M∈ T Mp

( )

be its tangent bundle. Then T M

( )

is also a differentiable manifold [1]. Let

1 n i

i i

X x

x =

∂

 

= _ _

∂

 

∑

and 1

n i i i dx

η=

∑

₌η be the expressions in local coordinates for the vector field X and the 1-form

η

_in_M_{. Let}

(

_{x y}i, i

)

_{be local coordinates}

of point in T M

( )

induced naturally from the coordinate chart

(

_{U x}, i

)

_in_M_.

The complete, vertical and horizontal lifts of tensor field have vital role in differential geometry of tangent bundle. In 2016, [2] studied Tachibana and Vishneveskii operators applied to vertical and horizontal lifts in almost paracontact structure on the tangent bundle T(M). The generalized almost r-contact structure in tangent bundle and integrability of structure is studied by the second author [3].

How to cite this paper: Das, L.S. and Khan, M.N.I. (2018) On Tachibana and Vishnevskii Operators Associated with Certain Structures in the Tangent Bundle. Journal of Applied Mathematics and Phys-ics, 6, 1968-1978.

https://doi.org/10.4236/jamp.2018.610168

Received: July 17, 2018 Accepted: October 7, 2018 Published: October 10, 2018

http://creativecommons.org/licenses/by/4.0/

(2)

DOI: 10.4236/jamp.2018.610168 1969 Journal of Applied Mathematics and Physics

This paper is organized as follows: Section 2 describes some basic definitions and notations. Section 3 deals with the study of Tachibana and Vishnevskii operators for generalized almost r-contact structure in tangent bundle.

2. Preliminaries

2.1. Vertical Lifts

If f is a function in M, we write _fV_{for the function in}_{T M}

( )

_{obtained by}

forming the composition of

π

:T M

( )

→M _and f M: →R_{, so that}

V

f = f

π

₍₁₎

where



is composition of f and pi. Thus, if a point _p__

_π

−1

( )

_U

has induced coordinates

(

_{x y}h, h

)

_then

( )

,

( )

V V

f p = f x y = f 

π

p = f p = f x ₍₂₎

Thus the value of _fV

( )

_p_ _{is constant along each fibre}

( )

p

T M and equal to the value f p

( )

. We call _fV_{the vertical lift of the function}_f_.

Vertical lifts to a unique algebraic isomorphism of the tensor algebra ℑ

( )

M

into the tensor algebra ℑ

(

T M

( )

)

with respect to constant coefficients by the conditions (Tensor product of P and Q)

(

_{P Q}_⊗

)

V ₌_PV_⊗_{Q P R}V,

(

₊

)

V₌_PV₊_RV

(3)

P, Q and R being arbitrary elements of ℑ

( )

M _.

Furthermore, the vertical lifts of tensor fields obey the general properties [4] [5]:

(a)

(

_{f g}_⋅

)

V ₌ _{f g}V V,

(

_{f g}₊

)

V ₌ _fV ₊_gV,

(b)

(

_{X Y}₊

)

V ₌_XV ₊_YV,

(

_{f X}_⋅

)

V ₌ _{f X X f}V V, V V ₌0,__{X Y}V, V_₌0,

 

(c)

(

_f _⋅_η

)

V ₌ _fV V_{η η}, V

( )

_XV ₌0,_{X Y}V

( )

V ₌0,

( )

0 1 0

0 0 1

, , , , .

f g M X Y M

η

M

∀ ∈ ℑ ∈ ℑ ∈ ℑ

2.2. Complete Lifts

If f is a function in M, we write _fC_{for the function in} _{T M}

( )

_{defined by}_[1]

( )

C

f =i df

and call _fC_{the complete lift of the function}_f_{. The complete lift} _fC_{of a}

function f has the local expression

C i i

f = ∂ = ∂y f f

with respect to the induced coordinates in T M

( )

, where ∂f _denotes i i

y f∂ _.

Suppose that 1

( )

0

X∈ ℑ M . We define a vector field _XC_in_{T M}

( )

_by

( )

C C C

X f = Xf

f being an arbitrary function in M and call _XC_{the complete lift of}_X_in

( )

(3)

The complete lift _XC_of_X_{with components} _xh_in_M_{has components}

: h

C h

x X

x

 

_∂ 

 

( )

. Suppose that 1

( )

0 M

η

∈ ℑ _{Then a 1-form}

_η

C_in_{T M}

( )

_{defined by}

( )

(

( )

)

C C _XC _X

η = η

X being an arbitrary vector field in M. We call

_η

C_{the complete lift of}

_η

_.

The complete lifts to a unique algebra isomorphism of the tensor algebra

( )

M

ℑ into the tensor algebra ℑ

(

T M

( )

)

with respect to constant coefficients, is given by the conditions

(

_{P Q}_⊗

)

C₌_PC_⊗_QV ₊_PV_⊗_QC,

(

_{P R}₊

)

C ₌_PC₊_RC P, Q and R being arbitrary elements of ℑ

( )

M _.

Moreover, the complete lifts of tensor fields obey the general properties [1] [4]:

(a)

( )

_fX C ₌ _{f X}C V ₊ _{f X}V C ₌

( )

_Xf C,_{X f}C V ₌

( )

_Xf V,_{X f}V C ₌

( )

_Xf V,

(b) _φV_XC₌

( )

_φ_X V,_φC_XV ₌

( ) ( )

_φ_X V, _φ_X C ₌_φC_XC,

(c) _ηV_XC ₌

(

_η

( )

_X

)

C,_ηC_XV ₌

(

_η

( )

_X

)

V,

(d) __{X Y}V, C_₌

[

_{X Y},

]

C,_IC ₌_{I I I}, V C ₌_XV,__{X Y}C, C_₌

[

_{X Y},

]

C

   

( )

0 1 0

0 0 1

, , , , .

f g M X Y M

η

M

∀ ∈ ℑ ∈ ℑ ∈ ℑ

2.3. Horizontal Lifts

The horizontal lift _fH_of 0

( )

0

f ∈ ℑ M to the tangent bundle T M

( )

by

( )

H _C

f = f − ∇γf

₍₄₎

where

( )

,

f f

γ

∇ = ∇

Let 1

( )

0

X∈ ℑ M . Then the horizontal lift _XH_of_X_{defined by}

H C

X =X − ∇γX ₍₅₎

in T M

( )

, where

( )

X X

γ

∇ = ∇

The horizontal lift _XH_of_X_{has the components} h

h i i

x x

 

_−Γ 

 (6)

( )

, where h j h i y ji

Γ = Γ .

The horizontal lift _SH_{of a tensor field}_S_{of arbitrary type in}_M_to_{T M}

( )

_is

defined by

H C

(4)

for all P Q, ∈ ℑ

( )

M . We have

(

_{P Q}

)

( )

_P _QV _PV

(

_Q

)

γ γ γ

∇ ⊗ = ∇ ⊗ + ⊗ ∇

or

(

_{P Q}_⊗

)

H ₌_PH_⊗_QV₊_PV_⊗_QH.

(8) In addition, the horizontal lifts of tensor fields obey the general properties [4] [6]:

(a) _{X f}H V

( )

_Xf V, V_XH

( )

_X V, C_XH

( )

_X H

( )

_XH;

γ

φ φ φ φ φ

= = = + ∇

(b) _ηV

( )

_XH ₌

(

_η

( )

_X

)

H,_ηC

( )

_XH ₌

(

_η

( )

_X

)

C₋_{γ η}

(

_

(

_∇_X

)

;

(c) H

( )

_XC H

(

_X

)

, H

( )

_XH 0

γ

η =η ∇ η =

( )

0 1 0 1

0 0 1 1

, , , , , .

f g M X Y M

η

M

φ

M

∀ ∈ ℑ ∈ ℑ ∈ ℑ ∈ ℑ

Let X be a vector field in an n-dimensional differentiable manifold M. The differential transformation LX is called Lie derivative with respect to X if

(a) 0

( )

0

, ,

X

L f Xf f= ∀ ∈ ℑ M

(b)

[

]

1

( )

0

, , , .

X

L Y= X Y ∀X Y∈ ℑ M

The Lie derivative L FX of a tensor field F of type (1, 1) with respect to a

vector field X is defined by

(

L FX

)

=

[

X FY F X Y,

] [

− ,

]

₍₉₎

where

[ ]

, is Lie bracket [1] page 113.

Let M be an n-dimensional differentiable manifold. Differential transformation of algebra T M

( )

defined by

( )

1

( )

0

: , ,

X

D= ∇ T M →T M X∈ ℑ M

(10) is called as covariant derivation with respect to vector field X if

(a) ∇fX gY+ t= ∇ + ∇f Xt g tY ,

(b) ∇Xf =Xf,

( )

0 1

0 0

, , , , .

f g M X Y M t M

∀ ∈ ℑ ∀ ∈ ℑ ∀ ∈ ℑ

and a transformation defined by

( )

1 1 1

0 0 0

: M M M

∇ ℑ × ℑ → ℑ ₍₁₁₎

is called affine connection [1].

Proposition 1. For any 1

( )

0 ,

X Y∈ ℑ M _[4]

(a) V, H

[

,

] (

V

)

V

( )

ˆ V,

X X

X Y X Y Y Y

  = − ∇ = − ∇

 

(b) C, H

[

,

]

H

(

)

, X

X Y X Y γ L Y

  = −

 

(c) H, V

[

,

]

V

(

)

V, Y

X Y X Y X

  = + ∇

 

(d) __{X Y}C, H_{ =}

[

_{X Y},

]

H₋_γ_{R X Y}ˆ

(

,

)

 

where Rˆ denotes the curvature tensor of the affine connection ∇ˆ _.

Proposition 2. For any 1

( )

0

( )

0 0

, ,

(5)

DOI: 10.4236/jamp.2018.610168 1972 Journal of Applied Mathematics and Physics horizontal lift of the affine connection ∇_to T M

( )

[1]

(a) ∇H_XVYV =0,

(b) ∇H_XVYH =0,

(c) H

(

)

,

V H V

X

X Y Y

∇ = ∇

(d) H

(

)

.

H H H

X

X Y Y

∇ = ∇

3. Tachibana and Vishnevskii Operators for Generalized

Almost R-Contact Structure in Tangent Bundle

Let M be a differentiable manifold of C∞_{class. Suppose that there are given a}

tensor field

φ

_{of type (1, 1), a vector field} ξ_p_{and a 1-form}ηp,p=1, 2, , r

satisfying [7] [8] [9]

(a) 2 2

1 r

p p p

a I

φ = +_

∑

₌ξ ⊗η

(b) φξ =_p 0

(c) η φ =p 0

(d)

η ξ

p

( )

q = −a2

δ

pq

 (12)

where p=1,2, , r and δpq denote the Kronecker delta while a and



are

non-zero complex numbers. The manifold M is called a generalized almost r-contact manifold with a generalized almost r-contact structure or in short with

(

φ η ξ, , , ,p p a

)

-structure.

Let us suppose that the base space M admits the Lorentzian almost r-para-contact structure. Then there exists a tensor field

φ

_{of type (1, 1),}

( )

r C∞ _{vector fields}

1, , ,2 p

ξ ξ  ξ and r C

( )

∞ _1-forms

1, , ,2 p

η η η such that Equation (12) are satisfied. Taking complete lifts of Equation (12) we obtain the following:

(a)

( )

2 2

{

}

1

r

H V H H V

p p p p

p

a I

φ = +

∑

₌ ξ ⊗η +ξ ⊗η

(b) H V 0, H C 0

p p

φ ξ

=

φ ξ

=

(c) V H 0, H V 0, H H 0, V V 0

p p p p

η φ

 =

η



φ

=

η



φ

=

η φ

 =

(d) H

( )

H V

( )

V 0, H

( )

V V

( )

H 2

p p p p p p p p a pq

η ξ

=

η ξ

=

η ξ

=

η ξ

= −

δ

 (13)

Let us define an element J of 1

( )

0

J T M by

(

)

1 r

H V V H H

p p p p

p

J

a

φ ξ η ξ η

=

= +

∑

⊗ + ⊗

 

(14) then in the view of Equation (13), it is easily shown that

2 V 2 V_, 2 H 2 H

J X =a X J X =a X

(6)

(a) H

( )

H r ₁

{

(

( )

)

V V

}

p p

p

JX X X

a

φ

₌

η

ξ

= +

∑

 

(b) V

( )

V r ₁

{

(

( )

)

V H

}

p p

p

JX X X

a

φ

₌

η

ξ

= +

∑

 

(15) for all 1

( )

0

X∈ ℑ M .

3.1. Tachibana Operator

Let

φ

_{be a tensor fieldof type (1, 1)}_i.e. 1

_{( )}

1 M

φ

∈ ℑ _and

( )

_{, 0} s

( )

r r s

M M

φ ∞

=

∈ ℑ =

∑

ℑ be a tensor algebra over R. A map Φ_{φ + >}_{r s} ₀ is called a Tachibana operator or Φφ operator on M if [2] [11]

(a) Φφ is linear with respect to constant coefficient,

(b) Φ ℑφ: ∗

( )

M → ℑrs+1

( )

M for all r and s,

(c)

(

_K C_L

)

(

_K

)

_{L K} _L

φ φ φ

Φ ⊗ = Φ ⊗ + ⊗ Φ for all K L, _{∈ ℑ}∗

( )

M _,

(d) ΦφXY = −

( )

LY

φ

X for all X Y, ∈ ℑ10

( )

M

where LY is Lie derivation with respect to Y,

(e)

( )

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

( ( Y ( ( Y Y

Y X Y

Y d X d X L X

X X X L X

φη

φ

η η η φ

Φ = ℑ Φ − ℑ Φ +

= Φ ℑ Φ − ℑ +



(16)

for all 0

( )

1 M

η

∈ ℑ _and 1

( )

0 ,

X Y∈ ℑ M _{, where}

( )

C Y

η η

X

η

Y

ℑ = = ⊗ _,

( )

s r M

∗

ℑ the module of pure tensor fields of type

( )

r s, on M with respect to the affinor field

ϕ

_{[12] [13]}_.

Theorem 3. For Tachibana operator on M L, X the operator Lie derivation with respect to 1

(

( )

)

1 ,

X J∈ ℑ T M defined by

(

)

1 r

H V V H H

p p p p

p

J

a

φ

₌

ξ

η

ξ

η

= +

∑

⊗ + ⊗

  _and

η

_{( )}

_Y =₀_,_{we have}

(a) V

(

( )

ˆ

)

1

(

ˆ

)

V _r V

H H

X p X p p

JY X

φ

Y _a =

η

Y

ξ

Φ_ = − ∇ +

∑

∇

(b) H

(

)

ˆ

(

,

)

1

(

)

ˆ

(

,

)

V

H r

H V

X p X p p

JY X L

φ

Y

γ

R X Y

φ

_a = L

η

Y

ξ

J R X Y

γ

Φ_ = − + +

∑

− 

(c) Φ_JYVXV =0

(d)

(

)

(

)

(

)

(

)

(

)

1

H

V

V V r

V H

X X p X p p

JY

V

r H

X p p

p

X L Y Y L Y

a Y

a

φ φ η ξ

η ξ

=

Φ = − + ∇ −

+ ∇

∑





 (17)

where 1

( )

0 ,

X Y∈ ℑ M _{, a tensor field} 1

( )

1 M

φ

∈ ℑ _{, a vector field}

ξ

_{and a}

1-form 0

( )

1 M

η

∈ ℑ .

Proof. For 1

(

)

r

H V V H H

p p p p

p

J

a

φ

₌

ξ

η

ξ

η

= +

∑

⊗ + ⊗

(7)

DOI: 10.4236/jamp.2018.610168 1974 Journal of Applied Mathematics and Physics (a)

(

)

(

)

[

]

(

)

( )

(

[

]

(

)

[

]

(

)

(

)

(

[

]

(

)

( )

(

)

[

]

1 1 1

, since ,

, ,

V H H V H V H V X

JY X X X

r

H V H V V H H H V

p p p p

p V

V V

H H

X

r _V _V r _V _V

V V H H

p Y p p Y p

p p

V V

V

H H

Y Y

X L J Y L JY JL Y L Y X Y

X JY X Y

a

X Y X Y Y

X Y X X Y X

a a

X Y φ X X Y

φ ξ η ξ η

φ φ

η ξ η ξ

φ φ

=

= =

Φ = − = − − =

      = −_ _+_ + ⊗ + ⊗ __ _     = −_ _+ + ∇ + + ∇ + + ∇   = −_ _ ∇ + + ∇

∑

       

(

)

(

V

)

X

[

]

(

)

(

)

(

[

]

(

)

( )

(

)

(

)

(

( )

)

(

)

(

)

(

)

(

)

(

)

(

)

(

( )

)

(

)

1 1 1 1 1 1 , ,

ˆ ˆ ˆ

ˆ

ˆ ˆ

as .

r _V _V r _V _V

V V H H

p Y p p Y p

p p

r

V V V V _H

X X X X p p

p

r V r _V

H H

X p p p X p

p p

r

V V

H

X X p X X p p

p

X Y X X Y X

a a

Y Y Y L Y

a

Y L Y

a a

L Y L Y Y Y

a

η ξ η ξ

φ φ φ η ξ

η ξ η ξ

η η φ η ξ

= = = = = = + + ∇ + + ∇ = − ∇ − ∇ + ∇ + − ∇ − = − = − ∇ − ∇

∑

      (18) (b)

(

)

(

)

[

]

(

)

( )

[

]

(

)

1 1 1 since , , , , , , , ˆ

since , , , ,

H H H H H H H H X

JY X X X

r

H H H V V H H H H

p p p p

p

r r

H

H H H H V H H V H H H H

p p p p

p p

H H H

X JY X Y

a

X Y X Y X Y X Y

a a

X Y X Y R X Y

φ ξ η ξ η

φ φ η ξ η ξ

γ =

= =

Φ = − = − − =

      = −_ _+_ + ⊗ + ⊗ __ _           = −_ _+ _ _+ _ _ + _ _   = −   =

∑

       

(

)

(

)

(

)

(

)

(

)

1

ˆ _, r V ˆ _{, .}

H _V

X X p p

p

L Y R X Y L Y J R X Y

a

φ γ φ η ξ γ

= − + −

∑

−  (19) (c)

(

)

(

)

[

]

(

)

( )

(

)

( )

(

( )

)

1 1

since ,

, , , , 0

,

as 0 , , 0.

V V V V V V V V X

JY X X X

V V V V V V

r

V H V V H H V

p p p p

p

r

H V V V H

p p p p

p

X JY J X Y X Y

X Y

a

Y X Y X Y

a

φ ξ η ξ η

η ξ φ η ξ

=

Φ = − = − − =

      = −_ _+ _ _ _ _=     = −_ _ + ⊗ + ⊗ _ _           = = −_ _− _ _=  

∑

        (20) (d)

(

)

[

]

(

)

[

]

(

)

(

[

]

(

)

[

]

(

)

(

)

(

[

]

(

)

( )

(

)

1 1 1

, since ,

, ,

since ,

H V V H V H V H X

JY X X X

r

V H H V V H H V H

p p p p

p

V V H V V

X X

r _V _V r _V _V

V V H H

p X p p X p

p p

p X X p X p

X JY X Y

a

X Y Y X Y Y

a a

L Y L Y L Y

φ ξ η ξ η

φ φ φ

η ξ η ξ

η η η η

=

= =

Φ = − = − + =

      = −_ _+_ + ⊗ + ⊗ __ _   = − + ∇ + − ∇ + − ∇ + − ∇ = −

∑

       

( )

(

)

(

)

(

)

(

)

(

)

(

)

1 1 .

p X X p X p

r V r V

V V _H _H

X X X p p X p p

p p

Y Y Y

L Y Y L Y Y

a a

η η

φ φ η ξ η ξ

= =

∇ = ∇ − ∇

= − + ∇ −

∑

+

∑

∇

(8)

Corollary 1. If we put Y=ξp i.e. η ξpH

( )

pH =η ξVp

( )

Vp =0,

( )

2

H V V H

p p p p a

η ξ

=

η ξ

= −

 , then we have

(a) V ₁

( )

_p ˆ

(

,

)

(

( )

ˆ

)

(

ˆ

)

p

V V

H r

H H

p X p X p p p

p

Jξ X a = L Xξ a R Xγ ξ φ ξ η ξ ξ

Φ_ =

∑

− − ∇ + ∇

(b)

(

)

(

)

(

)

(

)

(

)

(

)

1

1 1

ˆ ˆ _,

ˆ _, ˆ _, _.

H p

V H r V

H H V

X p X p p p X p p p

J

r V V r H H

p p p p p p

p p

X a L R X L

a

R X R X

a a

ξ ξ φ ξ φ γ ξ η ξ ξ

η γ ξ ξ η γ ξ ξ

=

= =

Φ = ∇ − − −

− −

∑





 

(c) V

(

ˆ _p

)

p

V V

Jξ X a ξ X

Φ_ = − ∇

(d)

(

)

(

)

(

)

(

)

(

)

1

1 .

H p

V

V V r

V H

X p X p p X p p p

J

V

r H

X p p p

p

X L L

a

ξ φ ξ φ ξ η ξ ξ

η ξ ξ

=

Φ = − + ∇ −

+ ∇

∑





(22)

3.2. Vishnevskii Operator

Let ∇_{is a linear connection and}

φ

be a tensor field of type (1, 1) on M. If the condition (d) of Tachibana operator replace by

(d’) ΨφXY = ∇φXY− ∇φ XY,

₍₂₃₎

( )

1 0 ,

X Y M

∀ ∈ ℑ , is a mapping wih linear connection ∇ _{. A map}

( )

: M M

φ ∗

Ψ ℑ → ℑ _{, which satisfies conditions (a), (b), (c), (e) of Tachibana}

operator and the condition (d’), is called Vishnevskii operator on M [2] [14]. Theorem 4. For Ψφ Vishnevskii operator on M and ∇H the horizontal lift

of an affine connection ∇_{in M to}T M

( )

, 1

(

( )

)

1

J∈ ℑ T M defined by (14), we have

(a) V 1

(

( )

)

H r

H

p p

p

JX Y _a =

η

X ξ Y

Ψ_ =

∑

∇

(b)

(

( )

)

(

)

(

)

(

)

1

ˆ ˆ

H

V V r V

V H

Y X p p Y p

JX

V

r H

p Y p

p

Y X L X X

a L X

a

φ φ η ξ

η ξ

=

Ψ = ∇ − − ∇

+

∑





(c) V 1

(

( )

)

H

p V r

V H V

p p

JX Y _a =

η

X ξ Y

Ψ_ = 

∑

∇

(d)

(

( )

)

(

)

(

)

(

)

1

ˆ ˆ

H

H H r V

H V

Y X p p Y p

JX

V

r V

p Y p

p

Y X L X X

a L X

a

φ φ η ξ

η ξ

=

Ψ = ∇ − − ∇

+

∑





(24)

where 1

( )

0 ,

X Y∈ ℑ M , a tensor field 1

( )

1 M

φ

∈ ℑ , vector fields ξp and a

1-form 0

( )

1 , 1, ,

p M p r

η

∈ ℑ =  .

(9)

DOI: 10.4236/jamp.2018.610168 1976 Journal of Applied Mathematics and Physics (a)

(

)

(

)

( ) ( )

( )

_{( )}

( )

(

)

1 1 1 1 1 as 0 as 0 .

V V V

V

H r V V H H V

p p p p

p

V V

V r H

p p

p

V p

p

H H H H H

JX JX X

r

H H H V V H H H H

p p p p _X

X _p

a

H H H

X

X X

a

r V H _H _H

p _X

p

r H

p p

Y Y J Y

Y Y

a

Y Y

X Y Y

a

X Y

a

φ ξ η ξ η

φ η ξ

ξ _φ

ξ

φ ξ η ξ η

η η = =  ₊ _⊗ ₊ _⊗    =   + = = ∑ ∑

Ψ = ∇ − ∇

  = ∇ − + ⊗ + ⊗ ∇   = ∇ ∇ = = ∇ ∇ = = ∇

∑

        (25) (b)

(

)

(

)

( )

(

)

(

)

(

)

[

]

(

( )

[

]

)

( )

[

]

(

)

( )

(

)

(

)

1 1 1 1

ˆ _, ˆ _,

ˆ _,

ˆ

H H H

H

H r V V H H H

p p p p

p

H

V H H H V

JX JX X

r

H V H V V H H H V

p p p p _X

X _p

a

r

V V

H V H H H

X p X p

X

p

V _V _H V _V

Y Y

r _V _V

H H

p Y p

p

V _V

Y Y

Y Y J Y

Y Y

a

Y Y Y

a

X X Y X X Y

X X Y

a

X L X

a

φ ξ η ξ η

φ

φ ξ η ξ η

φ η ξ

φ φ φ

η ξ φ φ =  ₊ _⊗ ₊ _⊗    =   = = ∑

Ψ = ∇ − ∇

  = ∇ −_ + ⊗ + ⊗ _∇   = ∇ − ∇ − ∇ = ∇ + − ∇ + − ∇ + = ∇ − −

∑

       

₍

₎

₍

₎

1 1 ˆ

r _V r _V

H H

p Y p p Y p

p X ap L X

η ξ η ξ

= =

∇ +

∑



∑

(26)

(c)

(

)

(

)

( )

(

( )

)

( )

(

)

_{( )} 1 1 1

1 as 0.

V V V

V

H r V V H H V

p p p p

p

V H

p

H V

p

V H V H V

JX JX X

r

H V H V V H H H V

p p p p _X

X _p

a

r _V

H V H V

p

X _p

r _V

H V H V

p _X

p

Y Y J Y

Y Y

a

Y X Y

a

X Y Y

a

φ ξ η ξ η

ξ φ

φ ξ η ξ η

ε _η η =  ₊ _⊗ ₊ _⊗    =   = = ∑

Ψ = ∇ − ∇

  = ∇ −_ + ⊗ + ⊗ _∇   = ∇ + ∇ = ∇ ∇ =

∑

      (27) (d)

(

)

(

)

( )

(

)

(

)

(

)

[

]

(

( )

[

]

)

( )

[

]

(

)

( )

(

)

(

)

1 1 1 1

ˆ _, ˆ _,

ˆ _,

ˆ

H H H

H

H r V V H H H

p p p p

p

H

H H H H H

JX JX X

r

H H H V V H H H H

p p p p _X

X _p

a

r

H H

H H H V V

X p X p

X _p

H _H _H H _H

Y Y

r _H _H

V H

p Y p

p

H _H

Y Y

Y Y J Y

Y Y

a

Y Y Y

a

X X Y X X Y

X X Y

a

X L X

a

φ ξ η ξ η

φ

φ ξ η ξ η

φ η ξ

φ φ φ

η ξ φ φ =  ₊ _⊗ ₊ _⊗    =   = = ∑

Ψ = ∇ − ∇

  = ∇ −_ + ⊗ + ⊗ _∇   = ∇ − ∇ − ∇ = ∇ + − ∇ + − ∇ + = ∇ − −

∑

       

₍

₎

₍

₎

1 1 ˆ

r _V r _V

V V

p Y p p Y p

p X ap L X

η ξ η ξ

= =

∇ +

∑



∑

(28)

Corollary 2. If we put Y =ξp i.e. η ξpH

( )

pH =η ξVp

( )

Vp =0,

( )

2

H V V H

p p p p a pq

η ξ

=

η ξ

= −

δ

(10)

(a) V

(

)

p

H H

p Jξ Y a ξ Y

Ψ_ = − ∇

(b)

(

)

(

)

(

)

(

)

(

)

1

ˆ ˆ

H p

V V r V

V H H

Y p Y p p p Y p p

J

V

r H

p Y p p

p

Y L

a L

a

η ξ ξ

=

Ψ = − ∇ − + ∇

−

∑





(c) V

( )

_p

p

V V

Jξ Y a ξ Y

Ψ_ = − ∇

(d)

(

( )

)

(

)

(

)

(

)

(

)

1

ˆ ˆ

.

H p

H _H _r V

H V

Y p Y p p Y p p p

J

V

r V

Y p p p

p

Y L

a L

a

η ξ ξ

=

Ψ = ∇ − + ∇

−

∑





(29)

4. Conclusion

The generalized almost r-contact structure on Tachibana and Visknnevskii operators in tangent bundle are introduced. We deduced the theorems on Tachibana and Visknnevskii operators with respect to Lie derivative and lifting theory.

Acknowledgements

We thank the Editor and the referee for their comments.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1] Yano, K. and Ishihara, S. (1973) Tangent and Cotangent Bundles. Marcel Dekker. Inc. New York.

[2] Cayir, H. (2016) Tachibana and Vishnevskii Operators Applied to _XV_and _XH

in Almost Paracontact Structure on Tangent Bundle T M

( )

. New Trends in Ma-thematical Sciences, 4, 105-115. https://doi.org/10.20852/ntmsci.2016318821

[3] Khan, M.N.I. (2017) A Note on Certain Structures in the Tangent Bundle. Far East Journal of Mathematical Sciences, 101, 1947-1965.

[4] Omran, T., Sharffuddin, A. and Husain, S.I. (1984) Lift of Structures on Manifold. Publications De L’institut Mathmatique (Beograd)(N.S.), 36, 93-97.

[5] Khan, M.N.I. (2014) Lifts of Hypersurfaces with Quarter-Symmetric Semi-Metric Connection to Tangent Bundles. Afrika Matematika,25, 475-482.

https://doi.org/10.1007/s13370-013-0150-x

[6] Das, L.S. and Khan, M.N.I. (2005) Almost R-Contact Structures on the Tangent Bundle. Differential Geometry-Dynamical Systems, 7, 34-41.

[7] Das, L.S. and Nivas, R. (2006) On Certain Structures Defined on the Tangent Bun-dle. Rocky Mountain Journal of Mathematics, 36, 1857-1866.

https://doi.org/10.1216/rmjm/1181069348

(11)

Su-DOI: 10.4236/jamp.2018.610168 1978 Journal of Applied Mathematics and Physics

periore Di Pisa, 26, 97-115.

[9] Blair, D.E. (1976) Contact Manifolds in Riemannian Geometry. Vol. 509, Springer Verlag, New York. https://doi.org/10.1007/BFb0079307

[10] Das, L.S., Nivas, R. and Khan, M.N.I. (2009) On Submanifolds of Co-Dimension 2 Immersed in a Hsuquaternion Manifold. Acta Mathematica Academiae Paedagogi-cae Nyregyhziensis, 25, 129-135.

[11] Magden, A. (2004) On Applications of the Tachibana Operator. Applied Mathe-matics and Computation, 147, 45-55.

https://doi.org/10.1016/S0096-3003(02)00649-5

[12] Tekkoyun, M. (2006) On Lifts of Paracomplex Structures. Turkish Journal of Math, 30, 197-210.

[13] Khan, M.N.I. (2016) Lift of Semi-Symmetric Non-Metric Connection on a Khler Manifold. Afrika Matematika, 27, 345-352.

https://doi.org/10.1007/s13370-015-0350-7