Int. J. Anal. Appl. (2022), 20:56

On Complete, Horizontal and Vertical Lifts From a Manifold With fλ (6,4) Structure

to Its Cotangent Bundle

Manisha M. Kankarej1,∗, Jai Pratap Singh2

1Department of Mathematics and Statistics, Zayed University, Dubai, UAE
2B.S.N.V.P.G. College, Lucknow university, Lucknow, India

∗Corresponding author: manisha.kankarej@gmail.com

Abstract. Manifolds with fλ(6,4) structure was defined and studied in the past. Later the geometry of

tangent and cotangent bundles in a differentiable manifold with fλ(6,4) structure was studied. The aim

of the present paper is to study complete, horizontal and vertical lifts from a manifold with fλ (6,4)-

structure to its cotangent bundle.

1. Introduction

The research on the properties of tensorial structure on manifolds and its extension to tangent and

cotangent bundles is always gaining attraction from the researchers. Yano [12], [13], [14] introduced

the idea of horizontal and vertical lifts on the tangent bundles. Kim [6] studied properties of f mani-

fold. Dube [5], Upadhyay and Gupta [11] studied integrability conditions of f 2v+4 + f 2 = 0; f 6 = 0

and of type (1; 1) and F(K;−(K −2)) - structure satisfying FK − FK−2 = 0; (F 6= 0; I). Srivas-
tava [9], [10] studied complete lifts of (1,1) tensor field F satisfying structure Fν+1 − λ2Fν−1 = 0
and extended in Mn to cotangent bundle. Nivas and Saxena [8] studied horizontal and complete

lifts from a manifold with fλ(7,−1) structure to its cotangent bundles. Li and Krupka [7] dis-
cussed the properties of tangent bundles. Cayir [1], [2] and [3] studied lifts of Fν+1,λ2Fν−1 structure.

Received: Aug. 11, 2022.

2010 Mathematics Subject Classification. 15A72, 47B47, 53A45, 53C15.

Key words and phrases. horizontal lift; vertical lift; complete lift; cotangent bundle; Nijenhuis tensor.

https://doi.org/10.28924/2291-8639-20-2022-56
ISSN: 2291-8639

© 2022 the author(s).

https://doi.org/10.28924/2291-8639-20-2022-56


2 Int. J. Anal. Appl. (2022), 20:56

Let M be a differentiable manifold of class C∞ and let CTM denote the cotangent bundle of M.

Then CTM is also a differentiable manifold of class C∞ and dimension 2n. Throughout this paper we

shall use the following notations and conventions:

(i) The map n : CTM → M denotes the projection map of CTM onto M.
(ii) Suffixes a, b, c. . . .h, i, j. . . ..take value 1 to n and i = i +n. Suffixes A, B, C, . . . ., take the

value 1 to 2n.

(iii) JrB (M) denote the set of tensor fields of class C
∞ and type (r,s) on M. Similarly JrB (

CTM)

denotes the set of such tensor fields in CTM.

(iv) Vector fields in M are denoted by X, Y, Z. . . .and the Lie-derivative by LX.

(v) The Lie product of X, Y is denoted by [X, Y].

If A is a point in M and n−1 (A) is a fibre over A. Any point p ∈ n−1 (A) is the ordered pair (A,
PA), where p is 1-form in M and ‘PA’ is the value of p at A. Let U be a coordinate neighborhood in M

such that A ∈ U. Then U induces a coordinate neighbourhood n−1 (U) in CTM and p ∈ n−1 (U)
by [4].

2. Complete Lift of fλ(6,4) - Structure

Let M be an n – dimensional differentiable manifold of class C∞. Suppose there exists on M, a tensor

field f (6=0) of type (1,1) by [6] and [10] we have

f 6 − λ2f 4 =0 (2.1)

where λ is a complex number not equal to zero. In such a manifold M, let us put

l =
f 4

λ2
, m = I −

f 4

λ2
(2.2)

where I denote the unit tensor field. Then it can be easily shown that

l2 = l, m2 = m, l +m = I and l ∗m = m∗ l =0 (2.3)

Thus, the operators ‘l’ and ‘m’ when applied to the tangent space M at a point are complementary

projection operators. Hence there exist complementary distributions L∗ and M∗ corresponding to the

projection operators ‘l’ and ‘m’ respectively. If the rank of f is constant everywhere and equal to r,

the dimension of L∗ and M∗ are r and (n-r) respectively. Let us call such a structure on M as fλ (6,4)

- structure of rank r.

Let f hi be component of f at A in the coordinate neighbourhood U of M. Then the complete lift f
C

of f is also a tensor field of type (1,1) in CTM, where components f
A
B in π

−1 (U) are given by [4]



Int. J. Anal. Appl. (2022), 20:56 3

f
h
i = f

h
i ; f

h
i =0 ;

f
h
i = pa

(
∂f ah
∂x i
−

∂f ai
∂xh

)
; f

h
i = f

i
h

(2.4)

where (x1, x2,. . . ., xn ) are coordinates of A in U and pa has components ( p1, p2,. . . .,pn). Thus

we can, write

f C =
(
f
A
B

)
=

[
f hi 0

pa
(
∂if ah − ∂hf

a
i

)
f ih

]
(2.5)

where ∂i = ∂/∂x i.

If we put

∂if ah − ∂h f
a
i =2∂ [i f

a
h ] ,

Then we can write f
A
B as

f C =
(
f
A
B

)
=

[
f hi 0

2pa∂
[
if ah
]
f ih

]
(2.6)

Thus, we have

(f
C
)
2
=

[
f hi 0

2pa∂
[
if ah
]
f ih

] [
f ij 0

2pt∂
[
jf ti
]
f
j
i

]
Or

(f
C
)
2
=

[
f hi f

i
j 0

2paf
i
j ∂

[
if ah
]
+ 2ptf

i
h∂

[
jf ti
]
f
j
i
f
i

h

]
(2.7)

If we put

2paf
i
j ∂ [if

a
h ]+ 2ptf

i
h∂

[
jf ti
]
= Lhj (2.8)

(f
C
)
2
=

[
f hi f

i
j 0

Lhj f
j
i
f
i

h

]
(2.9)

Squaring again from [4] we get

(f
C
)
4
=

[
f hi f

i
j 0

Lhj f
j
i
f
i

h

][
f
j
k
f kl 0

Ljl f
l
kf
k

j

]
Or

(f
C
)
4
=

[
f hi f

i
j f
j
k
f kl 0

f
j
k
f kl Lhj + f

j
i
f
i

h
Ljl f

l
kf
k

j
f
j
i
f
i

h

]
(2.10)

Thus

(f
C
)
6
=

[
f hi f

i
j f
j
k
f kl 0

f
j
k
f kl Lhj + f

j
i
f
i

h
Ljl f

l
kf
k

j
f
j
i
f
i

h

] [
f lmf

m
n 0

Lln f
n
mf

m
l

]
(2.11)



4 Int. J. Anal. Appl. (2022), 20:56

Or

(f
C
)
6
=


 f hi f ij f jkf kl f lmf mn 0
f
j
k
f kl f

l
mf

m
n Lhj + f

j
i
f
i

h
f lmf

m
n Ljl + f

l
kf
k

j
f
j
i
f
i

h
Lln f

n
mf

m
l f

l
k
f
k

j
f
j
i
f
i

h




Putting again

f
j
k
f kl f

l
mf

m
n Lhj + f

j
i
f
i

h
f lmf

m
n Ljl + f

l
kf
k

j f
j
i
f
i

h
Lln (2.12)

= λ2{f pq f
q
n Lhp + f

p
r f

r
h Lpn}

Thus, in view of the equations (2.12) and also (2.1), the above equation (2.11) takes the form

(f
C
)
6
=


 λ2 f hpf pq f qr f rn 0
λ2
{
f
p
q f

q
n Lhp + f

p
r f

r
h Lpn

}
λ2f

n
r f
r

qf
q

p
f
p

h




= λ2


 f hp f pq f qr f rn 0
f
p
q f

q
n Lhp + f

p
r f

r
h Lpn f

n
r f

r
qf
q

p
f
p

h




Or (f C)
6
− λ2 (f C)4 =0

Hence the complete lift f C of f also has fλ (6,4) - structure in the cotangent bundle CTM. Thus, we

have

Theorem 2.1. In order that the complete lift f C of a (1,1) tensor field f admitting fλ (6,4)-structure

in M may have the similar structure in the cotangent bundle CTM, it is necessary and sufficient that

f
j
k
f kl f

l
mf

m
n Lhj + f

j
i
f
i

h
f lmf

m
n Ljl + f

l
kf
k

j f
j
i
f
i

h
Lln = λ

2
{
f pq f

q
n Lhp + f

p
r f

r
h Lpn

}
3. Nijenhuis Tensor of Complete Lift of f 6

The Nijenhuis tensor of (1,1) tensor field f on M is given by

Nf ,f (X,Y )= [f X, f Y ]− f [f X, Y ]− f [X, f Y ]+ f 2[X,Y ] (3.1)

Also, for the complete lift of f 6, the Nijenhuis tensor is given by

N
(f
6
)
c
,(f
6
)
c (Xc,Y c)= [(f

6
)
c
Xc , (f 6)

c
Y
c
] − (f 6)

c
[(f 6)

c
Xc , Y c]

− (f 6)
c
[Xc , (f 6)

c
Y
c
] +(f

6
)
c
(f 6)

c
[Xc , Y c]

(3.2)

In the view of the equation (2.1), the above equation takes the form

N
(f
6
)
c
,(f
6
)
c (Xc,Y c)= [(λ

2
f 4)

c
Xc , (λ

2
f 4)

c
Y
c
]

−(λ2f 4)
c
[
(λ
2
f 4)

c
Xc, Y c

]
−(λ2f 4)

c
[
Xc, (λ

2
f 4)

c
Y
c
]

+(λ
2
f 4)

c
(λ
2
f 4)

c
[Xc , Y c]

(3.3)



Int. J. Anal. Appl. (2022), 20:56 5

= λ4{[(f 4)c Xc , (f 4)cY c] − (f 4)c
[
(f 4)

c
Xc, Y c

]
−(f 4)c

[
Xc, (f 4)

c
Y
c
]
+(f 4)

c
(f 4)

c
[Xc , Y c]}

Also,

(f 4)
c
Xc = (f 4X)

c
+ν(LXf

4) (3.4)

where (νf ) has components

(νf )=
[
p0a f

a
i

]
(3.5)

In view of the equation (3.4), the equation (3.3) takes the form of a horizontal lift of fλ(6,4)

structure.

N(
(f
4
)
C
,(f
4
)
C
) (XC, Y C) = λ4 {[(f 4X)C, (f 4Y )C] +[ν (LXf 4) , (f 4Y )C] +[(f 4X)C,ν (LY f 4)]

+
[
ν
(
LXf

4
)
, ν
(
LY f

4
)]
− (f 4)

C
[
(f
4
X)

C
, Y C

]
− (f 4)

C
[
ν
(
LXf

4
)C
, Y C

]
− (f 4)

C
[
XC,(f

4
Y )

C
]
− (f 4)

C [
XC,ν

(
LY f

4
)]
+ (f

4
)
C
(f
4
)
C
[XC, Y C]}

(3.6)

Let us now suppose that

LXf
4 − LY f 4 =0 (3.7)

The equation (3.6) takes the form

N(
(f
4
)
C
)
,(f
4
)
C

(
XC, Y C

)
= λ4

{
[(f
4
X)

C
, (f

4
Y )

C
]
}− (f 4)

C
[
(f
4
)
C
, Y C

]
(3.8)

− (f 4)
C
[
XC,(f

4
Y )

C
]
+ (f

4
)
C
(f
4
)
C
[XC, Y C]

Suppose further that the (1,1) tensor field f satisfies

f 4 = λ2I (3.9)

Then in the view of the equation (3.8), the equation (3.7) takes the form of

N(
(f
4
)
C
)
,(f
4
)
C

(
XC, Y C

)
= λ8

{[
XC, Y C

]
−
[
XC, Y C

]
−
[
XC, Y C

]
+
[
XC, Y C

]}
=0.

Hence, we have

Theorem 3.1. The Nijenhuis tensor of the complete lift of f 6 vanishes if the Lie – derivatives of the

tensor field f 4 with respect to X and Y are both zero and the tensor field f 2 acts as GF- structure

operator on M.



6 Int. J. Anal. Appl. (2022), 20:56

4. Horizontal Lift of fλ (6,4)- Structure

Let f, g be the tensor fields of type (1, 1) of manifold M. If f H be the horizontal lift of f, we have

by [4] and [14]

f HgH + gHf H = (f g + gf )
H (4.1)

Equating f and g, we get

(f H)
2
= (f 2)

H
(4.2)

Squaring equation (4.2) on both sides we get,

(f H)
4
= (f 4)

H
(4.3)

Taking cube of (4.2) and using (4.2) itself and (4.3) we get,

(f H)
6
= (f 6)

H
(4.4)

Since f gives fλ(6,4) – structure on M, so

f 6 − λ2f 4 = 0

Taking horizontal lift in the above equation we get,

(f 6)
H − λ2(f 4)

H
= 0 (4.5)

In view of the equation (4.3) and (4.4), the above equation (4.5) takes the form

(f 6)
H − λ2(f H)

4
= 0

Thus, we have the following theorem:

Theorem 4.1. Let f be the tensor field of type (1, 1) admitting fλ(6,4) structure in M. Then the

horizontal lift f H of f also admits the similar structure in the cotangent bundle cTM.

5. Vertical Lift of fλ (6,4)- Structure

Let f, g be the tensor fields of type (1, 1) of manifold M. If f V be the vertical lift of f, we have

f V gV + gV f V = (f g + gf )
V (5.1)

Equating f and g, we get

(f V )
2
= (f 2)

V
(5.2)

Squaring equation (5.2) on both sides we get,

(f V )
4
= (f 4)

V
(5.3)

Taking cube of (5.2) and using (5.2) itself and (5.3) we get,

(f V )
6
= (f 6)

V
(5.4)



Int. J. Anal. Appl. (2022), 20:56 7

Since f gives fλ(6,4) – structure on M, so

f 6 − λ2f 4 = 0

Taking vertical lift in the above equation we get,

(f 6)
V − λ2(f 4)

V
= 0 (5.5)

In view of the equation (5.3) and (5.4), the above equation (5.5) takes the form

(f 6)
V − λ2(f V )

4
= 0

Thus, we have the following theorem:

Theorem 5.1. Let f be the tensor field of type (1, 1) admitting fλ(6,4) structure in M. Then the

vertical lift f V of f also admits the similar structure in the cotangent bundle cTM.

6. Conclusion

In this research, fλ(6,4) structure has been defined on an n-dimensional differentiable manifold of

class C∞. Further properties of complete, horizontal and vertical lifts of fλ(6,4) structure are defined

on its cotangent bundle. The necessary and sufficient conditions for cotangent bundles to have the

properties of M in complete, horizontal and vertical lifts are also discussed. Properties of Nijenhuis

tensor of complete lift of f 6 is also a part of this paper.

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi-

cation of this paper.

References

[1] H. Cayir, Tachibana and Vishnevskii Operators Applied to XV and XH in Almost Paracontact Structure on Tangent

Bundle T(M), New Trends Math. Sci. 4 (2016), 105-115. https://doi.org/10.20852/ntmsci.2016318821.

[2] H. Cayir, Lie Derivatives of Almost Contact Structure and Almost Paracontact Structure With Respect To Xv and

XH on Tangent Bundle T(M), Proc. Inst. Math. Mech. 42 (2016), 38-49.

[3] H. Cayir, Some Notes on Lifts of the ((ν+1),λ2(ν−1)) - Structure on Cotangent and Tangent Bundle, Commun.
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[4] L.S. Das, R. Nivas, V.N. Pathak, On Horizontal and Complete Lifts From a Manifold With fλ(7;1)-Structure to Its

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[5] K.K. Dube, On a Differentiable Structure Satisfying f 2v+4 + f 2 =0; f 6 =0 of Type (1; 1), Nepali Math. Sci. Rep.

17 (1998), 99-102.

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	1. Introduction
	2. Complete Lift of f(6,4) - Structure
	3. Nijenhuis Tensor of Complete Lift of f6
	4. Horizontal Lift of f(6,4)- Structure
	5.  Vertical Lift of f(6,4)- Structure
	6. Conclusion
	References