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 $T\left(M\right)={\displaystyle {\cup }_{p\in M}}{T}_p\left(M\right)$ be its tangent bundle. Then $T\left(M\right)$ is also a differentiable manifold [1] . Let $X={\displaystyle {\sum }_{i=1}^n}{x}^i\left(\frac{\partial }{\partial x^i}\right)$ and $\eta ={\displaystyle {\sum }_{i=1}^n}{\eta }^{i}d{x}^i$ be the expressions in local coordinates for the vector field X and the 1-form $\eta$ in M. Let $\left(x^i\mathrm{,}{y}^i\right)$ be local coordinates of point in $T\left(M\right)$ induced naturally from the coordinate chart $\left(U\mathrm{,}{x}^i\right)$ 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] .

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 $f^V$ for the function in $T\left(M\right)$ obtained by forming the composition of $\pi \mathrm{<mo>\; :\; </mo>}T\left(M\right)\to M$ and $f\mathrm{<mo>\; :\; </mo>}M\to R$ , so that

$f^V=f\circ \pi$ (1)

where $\circ$ is composition of f and pi.

Thus, if a point $\tilde{p}ϵ{\pi }^{-1}\left(U\right)$ has induced coordinates $\left(x^h\mathrm{,}{y}^h\right)$ then

$f^V\left(\tilde{p}\right)=f^V\left(x,y\right)=f\circ \pi \left(\tilde{p}\right)=f\left(p\right)=f\left(x\right)$ (2)

Thus the value of $f^V\left(\tilde{p}\right)$ is constant along each fibre $T_p\left(M\right)$ and equal to the value $f\left(p\right)$ . We call $f^V$ the vertical lift of the function f.

Vertical lifts to a unique algebraic isomorphism of the tensor algebra $\Im \left(M\right)$ into the tensor algebra $\Im \left(T\left(M\right)\right)$ with respect to constant coefficients by the conditions (Tensor product of P and Q)

${\left(P\otimes Q\right)}^V=P^V\otimes Q^V,{\left(P+R\right)}^V=P^V+R^V$ (3)

P, Q and R being arbitrary elements of $\Im \left(M\right)$ .

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

(a) ${\left(f\cdot g\right)}^V=f^V{g}^V,{\left(f+g\right)}^V=f^V+g^V,$

(b) ${\left(X+Y\right)}^V=X^V+Y^V,{\left(f\cdot X\right)}^V=f^V{X}^V,X^V{f}^V=0,\left[X^V,Y^V\right]=0,$

(c) ${\left(f\cdot \eta \right)}^V=f^V{\eta }^V,{\eta }^V\left(X^V\right)=0,X^V\left(Y^V\right)=0,$

$\forall f\mathrm{,}g\in {\Im }_0^0\left(M\right)\mathrm{,}X\mathrm{,}Y\in {\Im }_0^1\left(M\right)\mathrm{,}\eta \in {\Im }_1^0\left(M\right)\mathrm{.}$

2.2. Complete Lifts

If f is a function in M, we write $f^C$ for the function in $T\left(M\right)$ 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

$f^C=y^i{\partial }_{i}f=\partial f$

with respect to the induced coordinates in $T\left(M\right)$ , where $\partial f$ denotes $y^i{\partial }_{i}f$ .

Suppose that $X\in {\Im }_0^1\left(M\right)$ . We define a vector field $X^C$ in $T\left(M\right)$ by

$X^C{f}^C={\left(Xf\right)}^C$

f being an arbitrary function in M and call $X^C$ the complete lift of X in $T\left(M\right)$ .

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

$X^C\mathrm{<mo>\; :\; </mo>}\left[\begin{array}{c}{x}^h\\ \partial x^h\end{array}\right]$

with respect to the induced coordinates in $T\left(M\right)$ .

Suppose that $\eta \in {\Im }_0^1\left(M\right)$ Then a 1-form ${\eta }^C$ in $T\left(M\right)$ defined by

${\eta }^C\left(X^C\right)={\left(\eta \left(X\right)\right)}^C$

X being an arbitrary vector field in M. We call ${\eta }^C$ the complete lift of $\eta$ .

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

${\left(P\otimes Q\right)}^C=P^C\otimes Q^V+P^V\otimes Q^C,{\left(P+R\right)}^C=P^C+R^C$

P, Q and R being arbitrary elements of $\Im \left(M\right)$ .

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

(a) ${\left(fX\right)}^C=f^C{X}^V+f^V{X}^C={\left(Xf\right)}^C,X^C{f}^V={\left(Xf\right)}^V,X^V{f}^C={\left(Xf\right)}^V,$

(b) ${\varphi }^{V}XC={\left(\varphi X\right)}^V,{\varphi }^C{X}^V={\left(\varphi X\right)}^V,{\left(\varphi X\right)}^C={\varphi }^C{X}^C,$

(c) ${\eta }^V{X}^C={\left(\eta \left(X\right)\right)}^C,{\eta }^C{X}^V={\left(\eta \left(X\right)\right)}^V,$

(d) $\left[X^V,Y^C\right]={\left[X,Y\right]}^C,I^C=I,I^V{I}^C=X^V,\left[X^C,Y^C\right]={\left[X,Y\right]}^C$

$\forall f\mathrm{,}g\in {\Im }_0^0\left(M\right)\mathrm{,}X\mathrm{,}Y\in {\Im }_0^1\left(M\right)\mathrm{,}\eta \in {\Im }_1^0\left(M\right)\mathrm{.}$

2.3. Horizontal Lifts

The horizontal lift $f^H$ of $f\in {\Im }_0^0\left(M\right)$ to the tangent bundle $T\left(M\right)$ by

${\left(f\right)}^H=f^C-{\nabla }_{\gamma }f$ (4)

where

${\nabla }_{\gamma }f=\gamma \left(\nabla f\right),$

Let $X\in {\Im }_0^1\left(M\right)$ . Then the horizontal lift $X^H$ of X defined by

$X^H=X^C-{\nabla }_{\gamma }X$ (5)

in $T\left(M\right)$ , where

${\nabla }_{\gamma }X=\gamma (\nabla X)$

The horizontal lift $X^H$ of X has the components

$\left[\begin{array}{c}{x}^h\\ -{\Gamma }_i^h{x}^i\end{array}\right]$ (6)

with respect to the induced coordinates in $T\left(M\right)$ , where ${\Gamma }_i^h=y^j{\Gamma }_{ji}^h$ .

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

$S^H=S^C-{\nabla }_{\gamma }S$ (7)

for all $P\mathrm{,}Q\in \Im \left(M\right)$ . We have

${\nabla }_{\gamma }\left(P\otimes Q\right)=\left({\nabla }_{\gamma }P\right)\otimes Q^V+P^V\otimes \left({\nabla }_{\gamma }Q\right)$

or

${\left(P\otimes Q\right)}^H=P^H\otimes Q^V+P^V\otimes Q^H.$ (8)

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

(a) $X^H{f}^V={\left(Xf\right)}^V,{\varphi }^V{X}^H={\left(\varphi X\right)}^V,{\varphi }^C{X}^H={\left(\varphi X\right)}^H+\left({\nabla }_{\gamma }\varphi \right)X^H;$

(b) ${\eta }^V\left(X^H\right)={\left(\eta \left(X\right)\right)}^H,{\eta }^C\left(X^H\right)={\left(\eta \left(X\right)\right)}^C-\gamma \left(\eta \circ \left(\nabla X\right)\right);$

(c) ${\eta }^H\left(X^C\right)={\eta }^H\left({\nabla }_{\gamma }X\right),{\eta }^H\left(X^H\right)=0$

$\forall f\mathrm{,}g\in {\Im }_0^0\left(M\right)\mathrm{,}X\mathrm{,}Y\in {\Im }_0^1\left(M\right)\mathrm{,}\eta \in {\Im }_1^0\left(M\right)\mathrm{,}\varphi \in {\Im }_1^1\left(M\right)\mathrm{.}$

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) $L_{X}f=Xf,\forall f\in {\Im }_0^0\left(M\right),$

(b) $L_{X}Y=\left[X,Y\right],\forall X,Y\in {\Im }_0^1\left(M\right).$

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

$\left(L_{X}F\right)=\left[X,FY\right]-F\left[X,Y\right]$ (9)

where $\left[\mathrm{,}\right]$ is Lie bracket [1] page 113.

Let M be an n-dimensional differentiable manifold. Differential transformation of algebra $T\left(M\right)$ defined by

$D={\nabla }_X:T\left(M\right)\to T\left(M\right),X\in {\Im }_0^1\left(M\right),$ (10)

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

(a) ${\nabla }_{fX+gY}t=f{\nabla }_{X}t+g{\nabla }_{Y}t,$

(b) ${\nabla }_{X}f=Xf,$

$\forall f\mathrm{,}g\in {\Im }_0^0\left(M\right)\mathrm{,}\forall X\mathrm{,}Y\in {\Im }_0^1\left(M\right)\mathrm{,}\forall t\in \Im \left(M\right)\mathrm{.}$

and a transformation defined by

$\nabla \mathrm{<mo>\; :\; </mo>}{\Im }_0^1\left(M\right)\times {\Im }_0^1\left(M\right)\to {\Im }_0^1\left(M\right)$ (11)

is called affine connection [1] .

Proposition 1. For any $X\mathrm{,}Y\in {\Im }_0^1\left(M\right)$ [4]

(a) $\left[X^V,Y^H\right]={\left[X,Y\right]}^V-{\left({\nabla }_{X}Y\right)}^V=-{\left({\hat{\nabla }}_{X}Y\right)}^V,$

(b) $\left[X^C,Y^H\right]={\left[X,Y\right]}^H-\gamma \left(L_{X}Y\right),$

(c) $\left[X^H,Y^V\right]={\left[X,Y\right]}^V+{\left({\nabla }_{Y}X\right)}^V,$

(d) $\left[X^C,Y^H\right]={\left[X,Y\right]}^H-\gamma \hat{R}\left(X,Y\right)$

where $\hat{R}$ denotes the curvature tensor of the affine connection $\hat{\nabla }$ .

Proposition 2. For any $X\mathrm{,}Y\in {\Im }_0^1\left(M\right)\mathrm{,}f\in {\Im }_0^0\left(M\right)$ and ${\nabla }^H$ is the horizontal lift of the affine connection $\nabla$ to $T\left(M\right)$ [1]

(a) ${\nabla }_{X^V}^H{Y}^V=0,$

(b) ${\nabla }_{X^V}^H{Y}^H=0,$

(c) ${\nabla }_{X^H}^H{Y}^V={\left({\nabla }_{X}Y\right)}^V,$

(d) ${\nabla }_{X^H}^H{Y}^H={\left({\nabla }_{X}Y\right)}^H.$

3. Tachibana and Vishnevskii Operators for Generalized Almost R-Contact Structure in Tangent Bundle

Let M be a differentiable manifold of $C^{\infty }$ class. Suppose that there are given a tensor field $\varphi$ of type (1, 1), a vector field ${\xi }_p$ and a 1-form ${\eta }_p,p=1,2,\cdots ,r$ satisfying [7] [8] [9]

(a) ${\varphi }^2=a^{2}I+ϵ{\displaystyle {\sum }_{p=1}^r{\xi }_p}\otimes {\eta }_p$

(b) $\varphi {\xi }_p=0$

(c) ${\eta }_p\circ \varphi =0$

(d) ${\eta }_p\left({\xi }_q\right)=-\frac{a^2}{ϵ}{\delta }_{pq}$ (12)

where $p=1,2,\cdots ,r$ and ${\delta }_{pq}$ denote the Kronecker delta while a and $ϵ$ 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 $\left(\varphi \mathrm{,}{\eta }_p\mathrm{,}{\xi }_p\mathrm{,}a\mathrm{,}ϵ\right)$ -structure.

Let us suppose that the base space M admits the Lorentzian almost r-para-contact structure. Then there exists a tensor field $\varphi$ of type (1, 1), $r\left(C^{\infty }\right)$ vector fields ${\xi }_1\mathrm{,}{\xi }_2\mathrm{,}\cdots ,{\xi }_p$ and $r\left(C^{\infty }\right)$ 1-forms ${\eta }_1\mathrm{,}{\eta }_2\mathrm{,}\cdots ,{\eta }_p$ such that Equation (12) are satisfied. Taking complete lifts of Equation (12) we obtain the following:

(a) ${\left({\varphi }^H\right)}^2=a^{2}I+ϵ{\displaystyle {\sum }_{p=1}^r\left\{{\xi }_p^V\otimes {\eta }_p^H+{\xi }_p^H\otimes {\eta }_p^V\right\}}$

(b) ${\varphi }^H{\xi }_p^V=0,{\varphi }^H{\xi }_p^C=0$

(c) ${\eta }_p^V\circ {\varphi }^H=0,{\eta }_p^H\circ {\varphi }^V=0,{\eta }_p^H\circ {\varphi }^H=0,{\eta }_p^V\circ {\varphi }^V=0$

(d) ${\eta }_p^H\left({\xi }_p^H\right)={\eta }_p^V\left({\xi }_p^V\right)=0,{\eta }_p^H\left({\xi }_p^V\right)={\eta }_p^V\left({\xi }_p^H\right)=-\frac{a^2}{ϵ}{\delta }_{pq}$ (13)

Let us define an element $\tilde{J}$ of $J_0^{1}T\left(M\right)$ by

$\tilde{J}={\varphi }^H+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}\left({\xi }_p^V\otimes {\eta }_p^V+{\xi }_p^H\otimes {\eta }_p^H\right)$ (14)

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

${\tilde{J}}^2{X}^V=a^2{X}^V,{\tilde{J}}^2{X}^H=a^2{X}^H$

which givess that $\tilde{J}$ is GF structure in $T\left(M\right)$ [10] .

Now in view of the Equation (15), we have

(a) $\tilde{J}{X}^H={\left(\varphi X\right)}^H+\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r\left\{{\left({\eta }_p\left(X\right)\right)}^V{\xi }_p^V\right\}}$

(b) $\tilde{J}{X}^V={\left(\varphi X\right)}^V+\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r\left\{{\left({\eta }_p\left(X\right)\right)}^V{\xi }_p^H\right\}}$ (15)

for all $X\in {\Im }_0^1\left(M\right)$ .

3.1. Tachibana Operator

Let $\varphi$ be a tensor fieldof type (1, 1) i.e. $\varphi \in {\Im }_1^1\left(M\right)$ and $\varphi \in \Im \left(M\right)={\displaystyle {\sum }_{r,s=0}^{\infty }}{\Im }_r^s\left(M\right)$ be a tensor algebra over R. A map ${{\Phi }_{\varphi }|}_{r+s>0}$ is called a Tachibana operator or ${\Phi }_{\varphi }$ operator on M if [2] [11]

(a) ${\Phi }_{\varphi }$ is linear with respect to constant coefficient,

(b) ${\Phi }_{\varphi }:{\Im }^{\ast }\left(M\right)\to {\Im }_{s+1}^r\left(M\right)$ for all r and s,

(c) ${\Phi }_{\varphi }\left(K{\otimes }^{C}L\right)=\left({\Phi }_{\varphi }K\right)\otimes L+K\otimes {\Phi }_{\varphi }L$ for all $K,L\in {\Im }^{\ast }\left(M\right)$ ,

(d) ${\Phi }_{\varphi X}Y=-\left(L_Y\varphi \right)X$ for all $X\mathrm{,}Y\in {\Im }_0^1(M)$

where $L_Y$ is Lie derivation with respect to Y,

(e) $\begin{array}{c}\left({\Phi }_{\varphi \eta }\right)Y=\mathrm{<mo\; stretchy="false">\; (\; </mo>}d\mathrm{<mo\; stretchy="false">\; (\; </mo>}{\Im }_Y\eta \left(\Phi X\right)-\mathrm{<mo\; stretchy="false">\; (\; </mo>}d\mathrm{<mo\; stretchy="false">\; (\; </mo>}{\Im }_Y\left(\eta \circ \Phi \right)X+\eta \left(\left(L_Y\varphi \right)X\right)\\ =\left(\Phi X\left({\Im }_Y\eta \right)\right)\left(\Phi X\right)-X\left({\Im }_{\varphi X}\eta \right)+\eta \left(\left(L_Y\varphi \right)X\right)\end{array}$ (16)

for all $\eta \in {\Im }_1^0\left(M\right)$ and $X\mathrm{,}Y\in {\Im }_0^1\left(M\right)$ , where ${\Im }_Y\eta =\eta \left(X\right)=\eta {\otimes }^{C}Y$ , ${\Im }_r^{\ast s}\left(M\right)$ the module of pure tensor fields of type $\left(r\mathrm{,}s\right)$ on M with respect to the affinor field $\phi$ [12] [13] .

Theorem 3. For Tachibana operator on $M\mathrm{,}{L}_X$ the operator Lie derivation with respect to $X\mathrm{,}\tilde{J}\in {\Im }_1^1\left(T\left(M\right)\right)$ defined by $\tilde{J}={\varphi }^H+\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r}\left({\xi }_p^V\otimes {\eta }_p^V+{\xi }_p^H\otimes {\eta }_p^H\right)$ and $\eta \left(Y\right)=0$ , we have

(a) ${\Phi }_{\tilde{J}{Y}^V}{X}^H=-{\left(\left({\hat{\nabla }}_X\varphi \right)Y\right)}^V+\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r{\left(\left({\hat{\nabla }}_X{\eta }_p\right)Y\right)}^V{\xi }_p^H}$

(b) ${\Phi }_{\tilde{J}{Y}^H}{X}^H=-{\left(\left(L_X\varphi \right)Y\right)}^H+\gamma \hat{R}\left(X,\varphi Y\right)+\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r{\left(\left(L_X{\eta }_p\right)Y\right)}^V{\xi }_p^V}-\tilde{J}\gamma \hat{R}\left(X,Y\right)$

(c) ${\Phi }_{\tilde{J}{Y}^V}{X}^V=0$

(d) $\begin{array}{c}{\Phi }_{\tilde{J}{Y}^H}{X}^V=-{\left(\left(L_X\varphi \right)Y\right)}^V+{\left(\left({\nabla }_X\varphi \right)Y\right)}^V-\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r{\left(\left(L_X{\eta }_p\right)Y\right)}^V{\xi }_p^H}\\ +\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r{\left(\left({\nabla }_X{\eta }_p\right)Y\right)}^V{\xi }_p^H}\end{array}$ (17)

where $X\mathrm{,}Y\in {\Im }_0^1\left(M\right)$ , a tensor field $\varphi \in {\Im }_1^1\left(M\right)$ , a vector field $\xi$ and a 1-form $\eta \in {\Im }_1^0\left(M\right)$ .

Proof. For $\tilde{J}={\varphi }^H+\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r}\left({\xi }_p^V\otimes {\eta }_p^V+{\xi }_p^H\otimes {\eta }_p^H\right)$ and $\eta \left(Y\right)=0$ , we get

(a) $\begin{array}{l}{\Phi }_{\tilde{J}{Y}^V}{X}^H=-\left(L_{X^H}\tilde{J}\right)Y^V=-\left(L_{X^H}\tilde{J}{Y}^V-\tilde{J}{L}_{X^H}{Y}^V\right),\mathrm{<mi>\; </mi>\; <mi>\; </mi>}\text{since}{L}_{X}Y=\left[X,Y\right]\\ =-\left[X^H,\tilde{J}{Y}^V\right]+\left({\varphi }^H+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}\left({\xi }_p^V\otimes {\eta }_p^V+{\xi }_p^H\otimes {\eta }_p^H\right)\right)\left[X^H,Y^V\right]\\ =-\left[X^H,{\left(\varphi Y\right)}^V\right]+{\varphi }^H\left({\left[X,Y\right]}^V+{\left({\nabla }_{X}Y\right)}^V\right)\\ +\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\eta }_p^V\left({\left[X,Y\right]}^V+{\left({\nabla }_{Y}X\right)}^V\right){\xi }_p^V+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\eta }_p^H\left({\left[X,Y\right]}^V+{\left({\nabla }_{Y}X\right)}^V\right){\xi }_p^H\\ =-\left[X^H,{\left(\varphi Y\right)}^V\right]{\left({\nabla }_{\varphi Y}X\right)}^V+{\varphi }^H\left({\left[X,Y\right]}^V+{\left({\nabla }_{Y}X\right)}^V\right)\end{array}$

$\begin{array}{l}+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\eta }_p^V\left({\left[X,Y\right]}^V+{\left({\nabla }_{Y}X\right)}^V\right){\xi }_p^V+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\eta }_p^H\left({\left[X,Y\right]}^V+{\left({\nabla }_{Y}X\right)}^V\right){\xi }_p^H\\ =-{\left(\left({\hat{\nabla }}_X\varphi \right)Y\right)}^V-{\left(\varphi {\hat{\nabla }}_{X}Y\right)}^V+{\left(\varphi \left({\hat{\nabla }}_{X}Y\right)\right)}^V+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\left(\left(L_X{\eta }_p\right)Y\right)}^V{\xi }_p^H\\ -\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\left(\left({\hat{\nabla }}_X{\eta }_p\right)Y\right)}^V{\xi }_p^H-\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\left({\eta }_p\left(L_{X}Y\right)\right)}^V{\xi }_p^H\\ \text{as}\mathrm{<mi>\; </mi>\; <mi>\; </mi>}\eta \left(L_{X}Y\right)=-\left(L_X{\eta }_p\right)Y=-{\left(\left({\hat{\nabla }}_X\varphi \right)Y\right)}^V-\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\left(\left({\hat{\nabla }}_X{\eta }_p\right)Y\right)}^V{\xi }_p^H.\end{array}$ (18)

(b) $\begin{array}{l}{\Phi }_{\tilde{J}{Y}^H}{X}^H=-\left(L_{X^H}\tilde{J}\right)Y^H=-\left(L_{X^H}\tilde{J}{Y}^H-\tilde{J}{L}_{X^H}{Y}^H\right)\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\text{since}{L}_{X}Y=\left[X,Y\right]\\ =-\left[X^H,\tilde{J}{Y}^H\right]+\left({\varphi }^H+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}\left({\xi }_p^V\otimes {\eta }_p^V+{\xi }_p^H\otimes {\eta }_p^H\right)\right)\left[X^H,Y^H\right]\\ =-\left[X^H,{\left(\varphi Y\right)}^H\right]+{\varphi }^H\left[X^H,Y^H\right]+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\eta }_p^V\left[X^H,Y^H\right]{\xi }_p^V+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\eta }_p^H\left[X^H,Y^H\right]{\xi }_p^H\\ \text{since}\left[X^H,Y^H\right]={\left[X,Y\right]}^H-\gamma \hat{R}\left(X,Y\right),\\ =-{\left(\left(L_X\varphi \right)Y\right)}^H+\gamma \hat{R}\left(X,\varphi Y\right)-\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\left(\left(L_X{\eta }_p\right)Y\right)}^V{\xi }_p^V-\tilde{J}\gamma \hat{R}\left(X,Y\right).\end{array}$ (19)

(c) $\begin{array}{l}{\Phi }_{\tilde{J}{Y}^V}{X}^V=-\left(L_{X^V}\tilde{J}\right)Y^V=-\left(L_{X^V}\tilde{J}{Y}^V-\tilde{J}{L}_{X^V}{Y}^V\right)\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\text{since}\mathrm{<mi>\; </mi>\; <mi>\; </mi>}{L}_{X}Y=\left[X,Y\right]\\ =-\left[X^V,\tilde{J}{Y}^V\right]+\tilde{J}\left[X^V,Y^V\right],\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\left[X^V,Y^V\right]=0\\ =-\left[X^V,\left({\varphi }^H+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}\left({\xi }_p^V\otimes {\eta }_p^V+{\xi }_p^H\otimes {\eta }_p^H\right)\right)Y^V\right]\\ \mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\text{as}\mathrm{<mi>\; </mi>\; <mi>\; </mi>}{\left({\eta }_p\left(Y\right){\xi }_p\right)}^H=0=-\left[X^V,{\left(\varphi Y\right)}^V\right]-\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}\left[X^V,{\left({\eta }_p\left(Y\right){\xi }_p\right)}^H\right]=0.\end{array}$ (20)

(d) $\begin{array}{l}{\Phi }_{\tilde{J}{Y}^H}{X}^V=-\left(L_{X^V}\tilde{J}\right)Y^H=-L_{X^V}\tilde{J}{Y}^H+\tilde{J}{L}_{X^V}{Y}^H,\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\text{since}{L}_{X}Y=\left[X,Y\right]\\ =-\left[X^V,\tilde{J}{Y}^H\right]+\left({\varphi }^H+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}\left({\xi }_p^V\otimes {\eta }_p^V+{\xi }_p^H\otimes {\eta }_p^H\right)\right)\left[X^V,Y^H\right]\\ =-{\left[X,\varphi Y\right]}^V+{\left({\nabla }_X\varphi Y\right)}^V+{\varphi }^H\left({\left[X,Y\right]}^V-{\left({\nabla }_{X}Y\right)}^V\right)\\ +\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\eta }_p^V\left({\left[X,Y\right]}^V-{\left({\nabla }_{X}Y\right)}^V\right){\xi }_p^V+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\eta }_p^H\left({\left[X,Y\right]}^V-{\left({\nabla }_{X}Y\right)}^V\right){\xi }_p^H\\ \text{since}{\eta }_p{L}_{X}Y=L_X{\eta }_p\left(Y\right)-\left(L_X{\eta }_p\right)Y,\mathrm{<mi>\; </mi>\; <mi>\; </mi>}{\eta }_p{\nabla }_{X}Y={\nabla }_X{\eta }_p\left(Y\right)-\left({\nabla }_X{\eta }_p\right)Y\\ =-{\left(\left(L_X\varphi \right)Y\right)}^V+{\left(\left({\nabla }_X\varphi \right)Y\right)}^V-\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\left(\left(L_X{\eta }_p\right)Y\right)}^V{\xi }_p^H+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\left(\left({\nabla }_X{\eta }_p\right)Y\right)}^V{\xi }_p^H.\end{array}$ (21),

Corollary 1. If we put $Y={\xi }_p$ i.e. ${\eta }_p^H\left({\xi }_p^H\right)={\eta }_p^V\left({\xi }_p^V\right)=0$ , ${\eta }_p^H\left({\xi }_p^V\right)={\eta }_p^V\left({\xi }_p^H\right)=-\frac{a^2}{ϵ}$ , then we have

(a) ${\Phi }_{\tilde{J}{\xi }_p^V}{X}^H=a{\displaystyle {\sum }_{p=1}^r{\left(L_{{\xi }_p}X\right)}^H}-a\gamma \hat{R}\left(X,{\xi }_p\right)-{\left(\left({\hat{\nabla }}_X\varphi \right){\xi }_p\right)}^V+{\left(\left({\hat{\nabla }}_X{\eta }_p\right){\xi }_p\right)}^V{\xi }_p^H$

(b) $\begin{array}{c}{\Phi }_{\tilde{J}{\xi }_p^H}{X}^H=a{\left({\hat{\nabla }}_X{\xi }_p\right)}^V-{\left(\left(L_X\varphi \right){\xi }_p\right)}^H-{\varphi }^H\gamma \hat{R}\left(X,{\xi }_p\right)-\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r{\left(\left(L_X{\eta }_p\right){\xi }_p\right)}^V{\xi }_p^V}\\ -\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r{\eta }_p^V\gamma \hat{R}\left(X,{\xi }_p\right){\xi }_p^V}-\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r{\eta }_p^H\gamma \hat{R}\left(X,{\xi }_p\right){\xi }_p^H}.\end{array}$

(c) ${\Phi }_{\tilde{J}{\xi }_p^V}{X}^V=-a{\left({\hat{\nabla }}_{{\xi }_p}X\right)}^V$

(d) $\begin{array}{c}{\Phi }_{\tilde{J}{\xi }_p^H}{X}^V=-{\left(\left(L_X\varphi \right){\xi }_p\right)}^V+{\left(\left({\nabla }_X\varphi \right){\xi }_p\right)}^V-\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r{\left(\left(L_X{\eta }_p\right){\xi }_p\right)}^V{\xi }_p^H}\\ +\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r{\left(\left({\nabla }_X{\eta }_p\right){\xi }_p\right)}^V{\xi }_p^H}.\end{array}$ (22)

3.2. Vishnevskii Operator

Let $\nabla$ is a linear connection and $\varphi$ be a tensor field of type (1, 1) on M. If the condition (d) of Tachibana operator replace by

(d’) ${\Psi }_{\varphi X}Y={\nabla }_{\varphi X}Y-\varphi {\nabla }_{X}Y,$ (23)

$\forall X\mathrm{,}Y\in {\Im }_0^1\left(M\right)$ , is a mapping wih linear connection $\nabla$ . A map ${\Psi }_{\varphi }\mathrm{<mo>\; :\; </mo>}{\Im }^{\ast }\left(M\right)\to \Im \left(M\right)$ , 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 ${\Psi }_{\varphi }$ Vishnevskii operator on M and ${\nabla }^H$ the horizontal lift of an affine connection $\nabla$ in M to $T\left(M\right)$ , $\tilde{J}\in {\Im }_1^1\left(T\left(M\right)\right)$ defined by (14), we have

(a) ${\Psi }_{\tilde{J}{X}^V}{Y}^H=\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r{\left({\eta }_p\left(X\right){\nabla }_{\xi }{}_{p}Y\right)}^H}$

(b) $\begin{array}{c}{\Psi }_{\tilde{J}{X}^H}{Y}^V={\left(\left({\hat{\nabla }}_Y\varphi \right)X\right)}^V-{\left(\left(L_X\varphi \right)X\right)}^V-\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r{\left({\eta }_p{\hat{\nabla }}_{Y}X\right)}^V{\xi }_p^H}\\ +\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r{\left({\eta }_p{L}_{Y}X\right)}^V{\xi }_p^H}\end{array}$

(c) ${\Psi }_{\tilde{J}{X}^V}{Y}^V=\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r{\left({\eta }_p\left(X\right)\right)}^V{\nabla }_{{\xi }_p^H}^H{Y}^V}$

(d) $\begin{array}{c}{\Psi }_{\tilde{J}{X}^H}{Y}^H={\left(\left({\hat{\nabla }}_Y\varphi \right)X\right)}^H-{\left(\left(L_X\varphi \right)X\right)}^H-\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r{\left({\eta }_p{\hat{\nabla }}_{Y}X\right)}^V{\xi }_p^V}\\ +\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r{\left({\eta }_p{L}_{Y}X\right)}^V{\xi }_p^V}\end{array}$ (24)

where $X\mathrm{,}Y\in {\Im }_0^1\left(M\right)$ , a tensor field $\varphi \in {\Im }_1^1\left(M\right)$ , vector fields ${\xi }_p$ and a 1-form ${\eta }_p\in {\Im }_1^0\left(M\right),p=1,\cdots ,r$ .

Proof.

(a) $\begin{array}{l}{\Psi }_{\tilde{J}{X}^V}{Y}^H={\nabla }_{\tilde{J}{X}^V}^H{Y}^H-\tilde{J}{\nabla }_{X^V}^H{Y}^H\\ ={\nabla }_{\left({\varphi }^H+\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r\left({\xi }_p^V\otimes {\eta }_p^V+{\xi }_p^H\otimes {\eta }_p^H\right)}\right)X^V}^H{Y}^H-\left({\varphi }^H+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}\left({\xi }_p^V\otimes {\eta }_p^V+{\xi }_p^H\otimes {\eta }_p^H\right)\right){\nabla }_{X^V}^H{Y}^H\\ ={\nabla }_{{\left(\varphi X\right)}^V+\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r{\left({\eta }_{p}X\right)}^V{\xi }_p^H}}{Y}^H\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\text{as}{\nabla }_{X^V}^H{Y}^H=0\\ =\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\left({\eta }_{p}X\right)}^V{\left({\nabla }_{{\xi }_p}Y\right)}^H\mathrm{<mi>\; </mi>\; <mi>\; </mi>}\text{as}{\nabla }_{{\left(\varphi X\right)}^V}^H{Y}^H=0\\ =\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\left({\eta }_p\left(X\right){\nabla }_{{\xi }_p}Y\right)}^H.\end{array}$ (25)

(b) $\begin{array}{l}{\Psi }_{\tilde{J}{X}^H}{Y}^V={\nabla }_{\tilde{J}{X}^H}^H{Y}^H-\tilde{J}{\nabla }_{X^H}^H{Y}^V\\ ={\nabla }_{\left({\varphi }^H+\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r\left({\xi }_p^V\otimes {\eta }_p^V+{\xi }_p^H\otimes {\eta }_p^H\right)}\right)X^H}^H{Y}^V-\left({\varphi }^H+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}\left({\xi }_p^V\otimes {\eta }_p^V+{\xi }_p^H\otimes {\eta }_p^H\right)\right){\nabla }_{X^H}^H{Y}^V\\ ={\nabla }_{{\left(\varphi X\right)}^H}^H{Y}^V-{\varphi }^H{\left({\nabla }_{X}Y\right)}^V-\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\eta }_p^H{\left({\nabla }_{X}Y\right)}^V{\xi }_p^H\\ ={\left({\hat{\nabla }}_Y\varphi X\right)}^V+{\left[\varphi X,Y\right]}^V-{\varphi }^H\left({\left({\hat{\nabla }}_{Y}X\right)}^V+{\left[X,Y\right]}^V\right)\\ -\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\eta }_p^H\left({\left({\hat{\nabla }}_{Y}X\right)}^V+{\left[X,Y\right]}^V\right){\xi }_p^H\\ ={\left(\left({\hat{\nabla }}_Y\varphi \right)X\right)}^V-{\left(\left(L_Y\varphi \right)X\right)}^V-\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\left({\eta }_p{\hat{\nabla }}_{Y}X\right)}^V{\xi }_p^H+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\left({\eta }_p{L}_{Y}X\right)}^V{\xi }_p^H\end{array}$ (26)

(c) $\begin{array}{l}{\Psi }_{\tilde{J}{X}^V}{Y}^V={\nabla }_{\tilde{J}{X}^V}^H{Y}^V-\tilde{J}{\nabla }_{X^V}^H{Y}^V\\ ={\nabla }_{\left({\varphi }^H+\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r\left({\xi }_p^V\otimes {\eta }_p^V+{\xi }_p^H\otimes {\eta }_p^H\right)}\right)X^V}^H{Y}^V-\left({\varphi }^H+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}\left({\xi }_p^V\otimes {\eta }_p^V+{\xi }_p^H\otimes {\eta }_p^H\right)\right){\nabla }_{X^V}^H{Y}^V\\ ={\nabla }_{{\left(\varphi X\right)}^V}^H{Y}^V+\frac{\epsilon }{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\left({\eta }_p\left(X\right)\right)}^V{\nabla }_{{\xi }_p^H}^H{Y}^V\\ =\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\left({\eta }_p\left(X\right)\right)}^V{\nabla }_{{\xi }_p^H}^H{Y}^V\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\text{as}{\nabla }_{{\left(\varphi X\right)}^V}^H{Y}^V=0.\end{array}$ (27)

(d) $\begin{array}{l}{\Psi }_{\tilde{J}{X}^H}{Y}^H={\nabla }_{\tilde{J}{X}^H}^H{Y}^H-\tilde{J}{\nabla }_{X^H}^H{Y}^H\\ ={\nabla }_{\left({\varphi }^H+\frac{ϵ}{a}{\displaystyle {\sum }_{p=1}^r\left({\xi }_p^V\otimes {\eta }_p^V+{\xi }_p^H\otimes {\eta }_p^H\right)}\right)X^H}^H{Y}^H-\left({\varphi }^H+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}\left({\xi }_p^V\otimes {\eta }_p^V+{\xi }_p^H\otimes {\eta }_p^H\right)\right){\nabla }_{X^H}^H{Y}^H\\ ={\nabla }_{{\left(\varphi X\right)}^H}^H{Y}^H-{\varphi }^H{\left({\nabla }_{X}Y\right)}^H-\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\eta }_p^V{\left({\nabla }_{X}Y\right)}^H{\xi }_p^V\\ ={\left({\hat{\nabla }}_Y\varphi X\right)}^H+{\left[\varphi X,Y\right]}^H-{\varphi }^H\left({\left({\hat{\nabla }}_{Y}X\right)}^H+{\left[X,Y\right]}^H\right)\\ -\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\eta }_p^V\left({\left({\hat{\nabla }}_{Y}X\right)}^H+{\left[X,Y\right]}^H\right){\xi }_p^H\\ ={\left(\left({\hat{\nabla }}_Y\varphi \right)X\right)}^H-{\left(\left(L_Y\varphi \right)X\right)}^H-\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\left({\eta }_p{\hat{\nabla }}_{Y}X\right)}^V{\xi }_p^V+\frac{ϵ}{a}{\displaystyle \underset{p=1}{\overset{r}{\sum }}}{\left({\eta }_p{L}_{Y}X\right)}^V{\xi }_p^V\end{array}$ (28)