On Tachibana and Vishnevskii Operators Associated with Certain Structures in the Tangent Bundle ()
1. Introduction
Let M be an n-dimensional differentiable manifold and let
be its tangent bundle. Then
is also a differentiable manifold [1] . Let
and
be the expressions in local coordinates for the vector field X and the 1-form
in M. Let
be local coordinates of point in
induced naturally from the coordinate chart
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
for the function in
obtained by forming the composition of
and
, so that
(1)
where
is composition of f and pi.
Thus, if a point
has induced coordinates
then
(2)
Thus the value of
is constant along each fibre
and equal to the value
. We call
the vertical lift of the function f.
Vertical lifts to a unique algebraic isomorphism of the tensor algebra
into the tensor algebra
with respect to constant coefficients by the conditions (Tensor product of P and Q)
(3)
P, Q and R being arbitrary elements of
.
Furthermore, the vertical lifts of tensor fields obey the general properties [4] [5] :
(a)
(b)
(c)
2.2. Complete Lifts
If f is a function in M, we write
for the function in
defined by [1]
and call
the complete lift of the function f. The complete lift
of a function f has the local expression
with respect to the induced coordinates in
, where
denotes
.
Suppose that
. We define a vector field
in
by
f being an arbitrary function in M and call
the complete lift of X in
.
The complete lift
of X with components
in M has components
with respect to the induced coordinates in
.
Suppose that
Then a 1-form
in
defined by
X being an arbitrary vector field in M. We call
the complete lift of
.
The complete lifts to a unique algebra isomorphism of the tensor algebra
into the tensor algebra
with respect to constant coefficients, is given by the conditions
P, Q and R being arbitrary elements of
.
Moreover, the complete lifts of tensor fields obey the general properties [1] [4] :
(a)
(b)
(c)
(d)
2.3. Horizontal Lifts
The horizontal lift
of
to the tangent bundle
by
(4)
where
Let
. Then the horizontal lift
of X defined by
(5)
in
, where
The horizontal lift
of X has the components
(6)
with respect to the induced coordinates in
, where
.
The horizontal lift
of a tensor field S of arbitrary type in M to
is defined by
(7)
for all
. We have
or
(8)
In addition, the horizontal lifts of tensor fields obey the general properties [4] [6] :
(a)
(b)
(c)
Let X be a vector field in an n-dimensional differentiable manifold M. The differential transformation
is called Lie derivative with respect to X if
(a)
(b)
The Lie derivative
of a tensor field F of type (1, 1) with respect to a vector field X is defined by
(9)
where
is Lie bracket [1] page 113.
Let M be an n-dimensional differentiable manifold. Differential transformation of algebra
defined by
(10)
is called as covariant derivation with respect to vector field X if
(a)
(b)
and a transformation defined by
(11)
is called affine connection [1] .
Proposition 1. For any
[4]
(a)
(b)
(c)
(d)
where
denotes the curvature tensor of the affine connection
.
Proposition 2. For any
and
is the horizontal lift of the affine connection
to
[1]
(a)
(b)
(c)
(d)
3. Tachibana and Vishnevskii Operators for Generalized Almost R-Contact Structure in Tangent Bundle
Let M be a differentiable manifold of
class. Suppose that there are given a tensor field
of type (1, 1), a vector field
and a 1-form
satisfying [7] [8] [9]
(a)
(b)
(c)
(d)
(12)
where
and
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
-structure.
Let us suppose that the base space M admits the Lorentzian almost r-para-contact structure. Then there exists a tensor field
of type (1, 1),
vector fields
and
1-forms
such that Equation (12) are satisfied. Taking complete lifts of Equation (12) we obtain the following:
(a)
(b)
(c)
(d)
(13)
Let us define an element
of
by
(14)
then in the view of Equation (13), it is easily shown that
which givess that
is GF structure in
[10] .
Now in view of the Equation (15), we have
(a)
(b)
(15)
for all
.
3.1. Tachibana Operator
Let
be a tensor fieldof type (1, 1) i.e.
and
be a tensor algebra over R. A map
is called a Tachibana operator or
operator on M if [2] [11]
(a)
is linear with respect to constant coefficient,
(b)
for all r and s,
(c)
for all
,
(d)
for all
where
is Lie derivation with respect to Y,
(e)
(16)
for all
and
, where
,
the module of pure tensor fields of type
on M with respect to the affinor field
[12] [13] .
Theorem 3. For Tachibana operator on
the operator Lie derivation with respect to
defined by
and
, we have
(a)
(b)
(c)
(d)
(17)
where
, a tensor field
, a vector field
and a 1-form
.
Proof. For
and
, we get
(a)
(18)
(b)
(19)
(c)
(20)
(d)
(21),
Corollary 1. If we put
i.e.
,
, then we have
(a)
(b)
(c)
(d)
(22)
3.2. Vishnevskii Operator
Let
is a linear connection and
be a tensor field of type (1, 1) on M. If the condition (d) of Tachibana operator replace by
(d’)
(23)
, is a mapping wih linear connection
. A map
, 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
Vishnevskii operator on M and
the horizontal lift of an affine connection
in M to
,
defined by (14), we have
(a)
(b)
(c)
(d)
(24)
where
, a tensor field
, vector fields
and a 1-form
.
Proof.
(a)
(25)
(b)
(26)
(c)
(27)
(d)
(28)
Corollary 2. If we put
i.e.
,
, then we have
(a)
(b)
(c)
(d)
(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.