If f is a function in M, we write f V for the function in T ( M ) obtained by forming the composition of π : T ( M ) → M and f : M → R , so that

f V = f ∘ π (1)

where ∘ is composition of f and pi.

Thus, if a point p ˜ ϵ π − 1 ( U ) has induced coordinates ( x h , y h ) then

f V ( p ˜ ) = f V ( x , y ) = f ∘ π ( p ˜ ) = f ( p ) = f ( x ) (2)

Thus the value of f V ( p ˜ ) is constant along each fibre T p ( M ) and equal to the value f ( p ) . We call f V 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 = P V ⊗ Q V , ( P + R ) V = P V + R V (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 V g V , ( f + g ) V = f V + g V ,

(b) ( X + Y ) V = X V + Y V , ( f ⋅ X ) V = f V X V , X V f V = 0 , [ X V , Y V ] = 0 ,

(c) ( f ⋅ η ) V = f V η V , η V ( X V ) = 0 , X V ( Y V ) = 0 ,

∀ f , g ∈ ℑ 0 0 ( M ) , X , Y ∈ ℑ 0 1 ( M ) , η ∈ ℑ 1 0 ( M ) .

If f is a function in M, we write f C for the function in T ( M ) defined by [1]

f C = i (df)

and call f C the complete lift of the function f. The complete lift f C of a function f has the local expression

f C = y i ∂ i f = ∂ f

with respect to the induced coordinates in T ( M ) , where ∂ f denotes y i ∂ i f .

Suppose that X ∈ ℑ 0 1 ( M ) . We define a vector field X C in T ( M ) by

X C f C = ( X f ) C

f being an arbitrary function in M and call X C the complete lift of X in T ( M ) .

The complete lift X C of X with components x h in M has components

X C : [ x h ∂ x h ]

with respect to the induced coordinates in T ( M ) .

Suppose that η ∈ ℑ 0 1 ( M ) Then a 1-form η C in T ( M ) defined by

η C ( X C ) = ( η ( X ) ) C

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 ℑ ( M ) into the tensor algebra ℑ ( T ( M ) ) with respect to constant coefficients, is given by the conditions

( P ⊗ Q ) C = P C ⊗ Q V + P V ⊗ Q C , ( P + R ) C = P C + R C

P, Q and R being arbitrary elements of ℑ ( M ) .

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

(a) ( f X ) C = f C X V + f V X C = ( X f ) C , X C f V = ( X f ) V , X V f C = ( X f ) V ,

(b)   ϕ V X C = ( ϕ X ) V , ϕ C X V = ( ϕ X ) V , ( ϕ X ) C = ϕ C X C ,

(c) η V X C = ( η ( X ) ) C , η C X V = ( η ( X ) ) V ,

(d) [ X V , Y C ] = [ X , Y ] C , I C = I , I V I C = X V , [ X C , Y C ] = [ X , Y ] C

∀ f , g ∈ ℑ 0 0 ( M ) , X , Y ∈ ℑ 0 1 ( M ) , η ∈ ℑ 1 0 ( M ) .

The horizontal lift f H of f ∈ ℑ 0 0 ( M ) to the tangent bundle T ( M ) by

( f ) H = f C − ∇ γ f (4)

where

∇ γ f = γ ( ∇ f ) ,

Let X ∈ ℑ 0 1 ( M ) . Then the horizontal lift X H of X defined by

X H = X C − ∇ γ X (5)

in T ( M ) , where

∇ γ X = γ (∇X)

The horizontal lift X H of X has the components

[ x h − Γ i h x i ] (6)

with respect to the induced coordinates in T ( M ) , where Γ i h = y j Γ j i h .

The horizontal lift S H of a tensor field S of arbitrary type in M to T ( M ) is defined by

S H = S C − ∇ γ S (7)

for all P , Q ∈ ℑ ( M ) . We have

∇ γ ( P ⊗ Q ) = ( ∇ γ P ) ⊗ Q V + P V ⊗ ( ∇ γ Q )

or

( P ⊗ Q ) H = P H ⊗ Q V + P V ⊗ Q H . (8)

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

(a) X H f V = ( X f ) V , ϕ V X H = ( ϕ X ) V , ϕ C X H = ( ϕ X ) H + ( ∇ γ ϕ ) X H ;

(b) η V ( X H ) = ( η ( X ) ) H , η C ( X H ) = ( η ( X ) ) C − γ ( η ∘ ( ∇ X ) ) ;

(c) η H ( X C ) = η H ( ∇ γ X ) , η H ( X H ) = 0

∀ f , g ∈ ℑ 0 0 ( M ) , X , Y ∈ ℑ 0 1 ( M ) , η ∈ ℑ 1 0 ( M ) , ϕ ∈ ℑ 1 1 ( M ) .

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

(a) L X f = X f , ∀ f ∈ ℑ 0 0 ( M ) ,

(b) L X Y = [ X , Y ] , ∀ X , Y ∈ ℑ 0 1 ( M ) .

The Lie derivative L X F of a tensor field F of type (1, 1) with respect to a vector field X is defined by

( L X F ) = [ X , F Y ] − 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

D = ∇ X : T ( M ) → T ( M ) , X ∈ ℑ 0 1 ( M ) , (10)

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

(a) ∇ f X + g Y t = f ∇ X t + g ∇ Y t ,

(b) ∇ X f = X f ,

  ∀ f , g ∈ ℑ 0 0 ( M ) , ∀ X , Y ∈ ℑ 0 1 ( M ) , ∀ t ∈ ℑ ( M ) .

and a transformation defined by

∇ : ℑ 0 1 ( M ) × ℑ 0 1 ( M ) → ℑ 0 1 ( M ) (11)

is called affine connection [1] .

Proposition 1. For any X , Y ∈ ℑ 0 1 ( M ) [4]

(a) [ X V , Y H ] = [ X , Y ] V − ( ∇ X Y ) V = − ( ∇ ^ X Y ) V ,

(b) [ X C , Y H ] = [ X , Y ] H − γ ( L X Y ) ,

(c) [ X H , Y V ] = [ X , Y ] V + ( ∇ Y X ) V ,

(d) [ X C , Y H ] = [ X , Y ] H − γ R ^ ( X , Y )

where R ^ denotes the curvature tensor of the affine connection ∇ ^ .

Proposition 2. For any X , Y ∈ ℑ 0 1 ( M ) , f ∈ ℑ 0 0 ( M ) and ∇ H is the horizontal lift of the affine connection ∇ to T ( M ) [1]

(a) ∇ X V H Y V = 0 ,

(b) ∇ X V H Y H = 0 ,

(c) ∇ X H H Y V = ( ∇ X Y ) V ,

(d) ∇ X H H Y H = ( ∇ X Y ) H .

Let ϕ be a tensor fieldof type (1, 1) i.e. ϕ ∈ ℑ 1 1 ( M ) and ϕ ∈ ℑ ( M ) = ∑ r , s = 0 ∞ ℑ r s ( M ) be a tensor algebra over R. A map Φ ϕ | r + s > 0 is called a Tachibana operator or Φ ϕ operator on M if [2] [11]

(a) Φ ϕ is linear with respect to constant coefficient,

(b) Φ ϕ : ℑ ∗ ( M ) → ℑ s + 1 r ( M ) for all r and s,

(c) Φ ϕ ( K ⊗ C L ) = ( Φ ϕ K ) ⊗ L + K ⊗ Φ ϕ L for all K , L ∈ ℑ ∗ ( M ) ,

(d) Φ ϕ X Y = − ( L Y ϕ ) X for all X , Y ∈ ℑ 0 1 (M)

where L Y is Lie derivation with respect to Y,

(e) ( Φ ϕ η ) Y = ( d ( ℑ Y η ( Φ X ) − ( d ( ℑ Y ( η ∘ Φ ) X + η ( ( L Y ϕ ) X ) = ( Φ X ( ℑ Y η ) ) ( Φ X ) − X ( ℑ ϕ X η ) + η ( ( L Y ϕ ) X ) (16)

for all η ∈ ℑ 1 0 ( M ) and X , Y ∈ ℑ 0 1 ( M ) , where ℑ Y η = η ( X ) = η ⊗ C Y , ℑ r ∗ s ( M ) 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 X , J ˜ ∈ ℑ 1 1 ( T ( M ) ) defined by J ˜ = ϕ H + ϵ a ∑ p = 1 r ( ξ p V ⊗ η p V + ξ p H ⊗ η p H ) and η ( Y ) = 0 , we have

(a) Φ J ˜ Y V X H = − ( ( ∇ ^ X ϕ ) Y ) V + ϵ a ∑ p = 1 r ( ( ∇ ^ X η p ) Y ) V ξ p H

(b) Φ J ˜ Y H X H = − ( ( L X ϕ ) Y ) H + γ R ^ ( X , ϕ Y ) + ϵ a ∑ p = 1 r ( ( L X η p ) Y ) V ξ p V − J ˜ γ R ^ ( X , Y )

(c) Φ J ˜ Y V X V = 0

(d) Φ J ˜ Y H X V = − ( ( L X ϕ ) Y ) V + ( ( ∇ X ϕ ) Y ) V − ϵ a ∑ p = 1 r ( ( L X η p ) Y ) V ξ p H     + ϵ a ∑ p = 1 r ( ( ∇ X η p ) Y ) V ξ p H (17)

where X , Y ∈ ℑ 0 1 ( M ) , a tensor field ϕ ∈ ℑ 1 1 ( M ) , a vector field ξ and a 1-form η ∈ ℑ 1 0 ( M ) .

Proof. For J ˜ = ϕ H + ϵ a ∑ p = 1 r ( ξ p V ⊗ η p V + ξ p H ⊗ η p H ) and η ( Y ) = 0 , we get

(a) Φ J ˜ Y V X H = − ( L X H J ˜ ) Y V = − ( L X H J ˜ Y V − J ˜ L X H Y V ) , since   L X Y = [ X , Y ] = − [ X H , J ˜ Y V ] + ( ϕ H + ϵ a ∑ p = 1 r ( ξ p V ⊗ η p V + ξ p H ⊗ η p H ) ) [ X H , Y V ] = − [ X H , ( ϕ Y ) V ] + ϕ H ( [ X , Y ] V + ( ∇ X Y ) V )     + ϵ a ∑ p = 1 r   η p V ( [ X , Y ] V + ( ∇ Y X ) V ) ξ p V + ϵ a ∑ p = 1 r   η p H ( [ X , Y ] V + ( ∇ Y X ) V ) ξ p H = − [ X H , ( ϕ Y ) V ] ( ∇ ϕ Y X ) V + ϕ H ( [ X , Y ] V + ( ∇ Y X ) V )

      + ϵ a ∑ p = 1 r   η p V ( [ X , Y ] V + ( ∇ Y X ) V ) ξ p V + ϵ a ∑ p = 1 r   η p H ( [ X , Y ] V + ( ∇ Y X ) V ) ξ p H = − ( ( ∇ ^ X ϕ ) Y ) V − ( ϕ ∇ ^ X Y ) V + ( ϕ ( ∇ ^ X Y ) ) V + ϵ a ∑ p = 1 r ( ( L X η p ) Y ) V ξ p H       − ϵ a ∑ p = 1 r ( ( ∇ ^ X η p ) Y ) V ξ p H − ϵ a ∑ p = 1 r ( η p ( L X Y ) ) V ξ p H     as η ( L X Y ) = − ( L X η p ) Y = − ( ( ∇ ^ X ϕ ) Y ) V − ϵ a ∑ p = 1 r ( ( ∇ ^ X η p ) Y ) V ξ p H . (18)

(b) Φ J ˜ Y H X H = − ( L X H J ˜ ) Y H = − ( L X H J ˜ Y H − J ˜ L X H Y H ) since   L X Y = [ X , Y ] = − [ X H , J ˜ Y H ] + ( ϕ H + ϵ a ∑ p = 1 r ( ξ p V ⊗ η p V + ξ p H ⊗ η p H ) ) [ X H , Y H ] = − [ X H , ( ϕ Y ) H ] + ϕ H [ X H , Y H ] + ϵ a ∑ p = 1 r   η p V [ X H , Y H ] ξ p V + ϵ a ∑ p = 1 r   η p H [ X H , Y H ] ξ p H       since   [ X H , Y H ] = [ X , Y ] H − γ R ^ ( X , Y ) , = − ( ( L X ϕ ) Y ) H + γ R ^ ( X , ϕ Y ) − ϵ a ∑ p = 1 r ( ( L X η p ) Y ) V ξ p V − J ˜ γ R ^ ( X , Y ) . (19)

(c) Φ J ˜ Y V X V = − ( L X V J ˜ ) Y V = − ( L X V J ˜ Y V − J ˜ L X V Y V ) since L X Y = [ X , Y ] = − [ X V , J ˜ Y V ] + J ˜ [ X V , Y V ] , [ X V , Y V ] = 0 = − [ X V , ( ϕ H + ϵ a ∑ p = 1 r ( ξ p V ⊗ η p V + ξ p H ⊗ η p H ) ) Y V ] as ( η p ( Y ) ξ p ) H = 0 = − [ X V , ( ϕ Y ) V ] − ϵ a ∑ p = 1 r [ X V , ( η p ( Y ) ξ p ) H ] = 0. (20)

(d) Φ J ˜ Y H X V = − ( L X V J ˜ ) Y H = − L X V J ˜ Y H + J ˜ L X V Y H , since   L X Y = [ X , Y ] = − [ X V , J ˜ Y H ] + ( ϕ H + ϵ a ∑ p = 1 r ( ξ p V ⊗ η p V + ξ p H ⊗ η p H ) ) [ X V , Y H ] = − [ X , ϕ Y ] V + ( ∇ X ϕ Y ) V + ϕ H ( [ X , Y ] V − ( ∇ X Y ) V )       + ϵ a ∑ p = 1 r   η p V ( [ X , Y ] V − ( ∇ X Y ) V ) ξ p V + ϵ a ∑ p = 1 r   η p H ( [ X , Y ] V − ( ∇ X Y ) V ) ξ p H     since     η p L X Y = L X η p ( Y ) − ( L X η p ) Y , η p ∇ X Y = ∇ X η p ( Y ) − ( ∇ X η p ) Y = − ( ( L X ϕ ) Y ) V + ( ( ∇ X ϕ ) Y ) V − ϵ a ∑ p = 1 r ( ( L X η p ) Y ) V ξ p H + ϵ a ∑ p = 1 r ( ( ∇ X η p ) Y ) V ξ p H . (21),

Corollary 1. If we put   Y = ξ p i.e. η p H ( ξ p H ) = η p V ( ξ p V ) = 0 , η p H ( ξ p V ) = η p V ( ξ p H ) = − a 2 ϵ , then we have

(a) Φ J ˜ ξ p V X H = a ∑ p = 1 r ( L ξ p X ) H − a γ R ^ ( X , ξ p ) − ( ( ∇ ^ X ϕ ) ξ p ) V + ( ( ∇ ^ X η p ) ξ p ) V ξ p H

(b) Φ J ˜ ξ p H X H = a ( ∇ ^ X ξ p ) V − ( ( L X ϕ ) ξ p ) H − ϕ H γ R ^ ( X , ξ p ) − ϵ a ∑ p = 1 r ( ( L X η p ) ξ p ) V ξ p V     − ϵ a ∑ p = 1 r η p V γ R ^ ( X , ξ p ) ξ p V − ϵ a ∑ p = 1 r η p H γ R ^ ( X , ξ p ) ξ p H .

(c) Φ J ˜ ξ p V X V = − a ( ∇ ^ ξ p X ) V

(d) Φ J ˜ ξ p H X V = − ( ( L X ϕ ) ξ p ) V + ( ( ∇ X ϕ ) ξ p ) V − ϵ a ∑ p = 1 r ( ( L X η p ) ξ p ) V ξ p H     + ϵ a ∑ p = 1 r ( ( ∇ X η p ) ξ p ) V ξ p H . (22)

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’) Ψ ϕ X Y = ∇ ϕ X Y − ϕ ∇ X Y , (23)

∀ X , Y ∈ ℑ 0 1 ( M ) , is a mapping wih linear connection ∇ . A map Ψ ϕ : ℑ ∗ ( M ) → ℑ ( M ) , 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 Ψ ϕ Vishnevskii operator on M and ∇ H the horizontal lift of an affine connection ∇ in M to T ( M ) , J ˜ ∈ ℑ 1 1 ( T ( M ) ) defined by (14), we have

(a) Ψ J ˜ X V Y H = ϵ a ∑ p = 1 r ( η p ( X ) ∇ ξ p Y ) H

(b) Ψ J ˜ X H Y V = ( ( ∇ ^ Y ϕ ) X ) V − ( ( L X ϕ ) X ) V − ϵ a ∑ p = 1 r ( η p ∇ ^ Y X ) V ξ p H     + ϵ a ∑ p = 1 r ( η p L Y X ) V ξ p H

(c) Ψ J ˜ X V Y V = ϵ a ∑ p = 1 r ( η p ( X ) ) V ∇ ξ p H H Y V

(d) Ψ J ˜ X H Y H = ( ( ∇ ^ Y ϕ ) X ) H − ( ( L X ϕ ) X ) H − ϵ a ∑ p = 1 r ( η p ∇ ^ Y X ) V ξ p V     + ϵ a ∑ p = 1 r ( η p L Y X ) V ξ p V (24)

where X , Y ∈ ℑ 0 1 ( M ) , a tensor field ϕ ∈ ℑ 1 1 ( M ) , vector fields ξ p and a 1-form η p ∈ ℑ 1 0 ( M ) , p = 1 , ⋯ , r .

Proof.

(a) Ψ J ˜ X V Y H = ∇ J ˜ X V H Y H − J ˜ ∇ X V H Y H = ∇ ( ϕ H + ϵ a ∑ p = 1 r ( ξ p V ⊗ η p V + ξ p H ⊗ η p H ) ) X V H Y H − ( ϕ H + ϵ a ∑ p = 1 r ( ξ p V ⊗ η p V + ξ p H ⊗ η p H ) ) ∇ X V H Y H = ∇ ( ϕ X ) V + ϵ a ∑ p = 1 r ( η p X ) V ξ p H Y H as   ∇ X V H Y H = 0 = ϵ a ∑ p = 1 r ( η p X ) V ( ∇ ξ p Y ) H as   ∇ ( ϕ X ) V H Y H = 0 = ϵ a ∑ p = 1 r ( η p ( X ) ∇ ξ p Y ) H . (25)

(b) Ψ J ˜ X H Y V = ∇ J ˜ X H H Y H − J ˜ ∇ X H H Y V = ∇ ( ϕ H + ϵ a ∑ p = 1 r ( ξ p V ⊗ η p V + ξ p H ⊗ η p H ) ) X H H Y V − ( ϕ H + ϵ a ∑ p = 1 r ( ξ p V ⊗ η p V + ξ p H ⊗ η p H ) ) ∇ X H H Y V = ∇ ( ϕ X ) H H Y V − ϕ H ( ∇ X Y ) V − ϵ a ∑ p = 1 r   η p H ( ∇ X Y ) V ξ p H = ( ∇ ^ Y ϕ X ) V + [ ϕ X , Y ] V − ϕ H ( ( ∇ ^ Y X ) V + [ X , Y ] V )       − ϵ a ∑ p = 1 r   η p H ( ( ∇ ^ Y X ) V + [ X , Y ] V ) ξ p H = ( ( ∇ ^ Y ϕ ) X ) V − ( ( L Y ϕ ) X ) V − ϵ a ∑ p = 1 r ( η p ∇ ^ Y X ) V ξ p H + ϵ a ∑ p = 1 r ( η p L Y X ) V ξ p H (26)

(c) Ψ J ˜ X V Y V = ∇ J ˜ X V H Y V − J ˜ ∇ X V H Y V = ∇ ( ϕ H + ϵ a ∑ p = 1 r ( ξ p V ⊗ η p V + ξ p H ⊗ η p H ) ) X V H Y V − ( ϕ H + ϵ a ∑ p = 1 r ( ξ p V ⊗ η p V + ξ p H ⊗ η p H ) ) ∇ X V H Y V = ∇ ( ϕ X ) V H Y V + ε a ∑ p = 1 r ( η p ( X ) ) V ∇ ξ p H H Y V = ϵ a ∑ p = 1 r ( η p ( X ) ) V ∇ ξ p H H Y V as   ∇ ( ϕ X ) V H Y V = 0. (27)

(d) Ψ J ˜ X H Y H = ∇ J ˜ X H H Y H − J ˜ ∇ X H H Y H = ∇ ( ϕ H + ϵ a ∑ p = 1 r ( ξ p V ⊗ η p V + ξ p H ⊗ η p H ) ) X H H Y H − ( ϕ H + ϵ a ∑ p = 1 r ( ξ p V ⊗ η p V + ξ p H ⊗ η p H ) ) ∇ X H H Y H = ∇ ( ϕ X ) H H Y H − ϕ H ( ∇ X Y ) H − ϵ a ∑ p = 1 r   η p V ( ∇ X Y ) H ξ p V = ( ∇ ^ Y ϕ X ) H + [ ϕ X , Y ] H − ϕ H ( ( ∇ ^ Y X ) H + [ X , Y ] H )       − ϵ a ∑ p = 1 r   η p V ( ( ∇ ^ Y X ) H + [ X , Y ] H ) ξ p H = ( ( ∇ ^ Y ϕ ) X ) H − ( ( L Y ϕ ) X ) H − ϵ a ∑ p = 1 r ( η p ∇ ^ Y X ) V ξ p V + ϵ a ∑ p = 1 r ( η p L Y X ) V ξ p V (28)

Corollary 2. If we put Y = ξ p i.e. η p H ( ξ p H ) = η p V ( ξ p V ) = 0 , η p H ( ξ p V ) = η p V ( ξ p H ) = − a 2 ϵ δ p q , then we have

(a) Ψ J ˜ ξ p V Y H = − a ( ∇ ξ p Y ) H

(b) Ψ J ˜ ξ p H Y V = − ϕ H ( ∇ ^ Y ξ p ) V − ( ( L Y ϕ ) ξ p ) V + ϵ a ∑ p = 1 r ( η p ( ∇ ^ Y ξ p ) V ) ξ p H     − ϵ a ∑ p = 1 r ( η p ( L Y ξ p ) V ) ξ p H

(c) Ψ J ˜ ξ p V Y V = − a ( ∇ ξ p Y ) V

(d) Ψ J ˜ ξ p H Y H = ( ( ∇ ^ Y ϕ ) ξ p ) H − ( ( L Y ϕ ) ξ p ) H + ϵ a ∑ p = 1 r ( ( ∇ ^ Y η p ) ξ p ) V ξ p V     − ϵ a ∑ p = 1 r ( ( L Y η p ) ξ p ) V ξ p V . (29)