Konuralp Journal of Mathematics c Volume 4 No. 2 pp. 209–216 (2016) KJM
COVARIENT DERIVATIVES OF ALMOST CONTACT STRUCTURE AND ALMOST PARACONTACT STRUCTURE WITH RESPECT TO X C AND X V ON TANGENT BUNDLE T (M ) HAS ¸ IM C ¸ AYIR
Abstract. The differential geometry of tangent bundles was studied by several authors, for example: D. E. Blair [1], V. Oproiu [4], A. Salimov [5], Yano and Ishihara [8] and among others. It is well known that differant structures deffined on a manifold M can be lifted to the same type of structures on its tangent bundle. Several authors cited here in obtained result in this direction. Our goal is to study covarient derivatives of almost contact structure and almost paracontact structure with respect to X C and X V on tangent bundle T (M ). In addition, this covarient derivatives which obtained shall be studied for some special values in almost contact structure and almost paracontact structure.
1. Introduction Let M be an n−dimensional differentiable manifold of class C ∞ and let Tp (M ) be the tangent space of M at a point p of M . Then the set [8] (1.1)
T (M ) = ∪ Tp (M ) p∈M
is called the tangent bundle over the manifold M. For any point p˜ of T (M ), the correspondence p˜ → p determines the bundle projection π : T (M ) → M , Thus π(˜ p) = p, where π : T (M ) → M defines the bundle projection of T (M ) over M. The set π −1 (p) is called the fibre over p ∈ M and M the base space. Suppose that the base space M is covered by a system of coordinate neighbourhoods U ; xh , where (xh ) is a system of local coordinates defined in the neighbourhood U of M. The open set π −1 (U ) ⊂ T (M ) is naturally differentiably homeomorphic to the direct product U × Rn , Rn being the n−dimensional vector space over the real field R, in such a way that a point p˜ ∈ Tp (M )(p ∈ U ) is represented by an ordered pair (P, X) of the point p ∈ U , and a vector X ∈ Rn ,whose components are given by the cartesian coordinates (y h ) of p˜ in the tangent space Tp (M ) with respect 2000 Mathematics Subject Classification. 53C05; 53C15. Key words and phrases. Covarient Derivative,Almost Contact Structure,Almost Paracontact Structure. 209
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to the natural base {∂h }, where ∂h = ∂x∂ h . Denoting by (xh ) the coordinates of p = π(˜ p) in U and establishing the correspondence (xh , y h ) → p˜ ∈ π −1 (U ), we can introduce a system of local coordinates (xh , y h ) in the open set π −1 (U ) ⊂ T (M ). Here we cal (xh , y h ) the coordinates in π −1 (U ) induced from (xh ) or simply, the induced coordinates in π −1 (U ). We denote by =rs (M ) the set of all tensor fields of class C ∞ and of type (r, s) in ∞ P M . We now put =(M ) = =rs (M ), which is the set of all tensor fields in M . r,s=0
Similarly, we denote by =rs (T (M )) and =(T (M )) respectively the corresponding sets of tensor fields in the tangent bundle T (M ). 1.1. Vertical lifts. 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 (1.2)
f v = f oπ
Thus, if a point p˜ ∈ π −1 (U ) has induced coordinates (xh , y h ), then (1.3)
f v (˜ p) = f v (x, y) = f oπ(˜ p) = f (p) = f (x).
Thus the value of f v (˜ p) is constant along each fibre Tp (M ) and equal to the value f (p). We call f v the vertical lift of the function f [8]. ˜ ∈ =1 (T (M )) be such that Xf ˜ v = 0 for all f ∈ =0 (M ). Then we say Let X 0 0 ˜ h X ˜ is a vertical vector field. Let ˜ h¯ be components of X ˜ with respect to the that X X ˜ is vertical if and only if its components in π −1 (U ) induced coordinates. Then X satisfy ˜ h 0 X . = (1.4) ¯ h ˜ X h¯ X Suppose that X ∈ =10 (M ), so that is a vector field in M . We define a vector field X v in T (M ) by (1.5)
X v (ι ω) = (ωX)v
ω being an arbitrary 1−form in M. We cal X v the vertical lift of X [8]. Let ω ˜ ∈ =01 (T (M )) be such that ω ˜ (X)v = 0 for all X ∈ =10 (M ). Then we say that ω ˜ is a vertical 1−form in T (M ). We define the vertical lift ω v of the 1−form ω by (1.6)
ω v = (ωi )v (dxi )v
in each open set π −1 (U ), where (U ; xh ) is coordinate neighbourhood in M and ω is given by ω = ωi dxi in U . The vertical lift ω v of ω with lokal expression ω = ωi dxi has components of the form (1.7)
ω v : (ω i , 0)
with respect to the induced coordinates in T (M ). Vertical lifts to a unique algebraic isomorphism of the tensor algebra =(M ) into the tensor algebra =(T (M )) with respect to constant coefficients by the conditions (1.8)
(P ⊗ Q)V = P V ⊗ QV , (P + R)V = P V + RV
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P, Q and R being arbitrary elements of T (M ). The vertical lifts F V of an element F ∈ =11 (M ) with lokal components Fih has components of the form [8] 0 0 V (1.9) F : . Fih 0 Vertical lift has the following formulas [3, 8]: (1.10)
v
= f v X v , I v X v = 0, η v (X v ) = 0
v
= f v η v , [X v , Y v ] = 0, ϕv X v = 0
(f X) (f η)
v v
X f
=
0, X v f v = 0
hold good, where f ∈ =00 (Mn ), X, Y ∈ =10 (Mn ), η ∈ =01 (Mn ), ϕ ∈ =11 (Mn ), I = idMn . 1.2. Complete lifts. If f is a function in M , we write f c for the function in T (M ) defined by f c = ι(df ) and call f c the comple lift of the function f . The complete lift f c of a function f has the lokal expression (1.11)
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 ∈ =10 (M ). Then we define a vector field X c in T (M ) by (1.12)
X c f c = (Xf )c ,
f being an arbitrary function in M and call X c the complete lift of X in T (M ) [2, 8]. The complete lift X c of X with components xh in M has components h X (1.13) Xc = ∂X h with respect to the induced coordinates in T (M ). Suppose that ω ∈ =01 (M ), then a 1−form ω c in T (M ) defined by (1.14)
ω c (X c ) = (ωX)c
X being an arbitrary vector field in M . We call ω c the complete lift of ω. The complete lift ω c of ω with components ωi in M has components of the form (1.15)
ω c : (∂ωi, ωi )
with respect to the induced coordinates in T (M ) [2]. The complete lifts to a unique algebra isomorphism of the tensor algebra =(M ) into the tensor algebra =(T (M )) with respect to constant coefficients, is given by the conditions (1.16)
(P ⊗ Q)C = P C ⊗ QV + P V ⊗ QC , (P + R)C = P C + RC ,
where P, Q and R being arbitrary elements of T (M ). The complete lifts F C of an element F ∈ =11 (M ) with lokal components Fih has components of the form Fih 0 (1.17) FC : . ∂Fih Fih In addition, we know that the complete lifts are defined by [3, 8]:
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(1.18)
c
=
f c X v + f v X c = (Xf )c ,
X cf v
=
(Xf ) , η v (xc ) = (η (x)) ,
(f X)
Xvf c
v
v
v
v
=
(Xf ) , ϕv X c = (ϕX) ,
v
=
(ϕX) , (ϕX) = ϕc X c ,
η v (X c )
=
(η (X)) , η c (X v ) = (η (X)) ,
[X v , Y c ]
=
[X, Y ] , I c = I, I v X c = X v , [X c , Y c ] = [X, Y ] .
c
ϕ X
v
c
c
v
v
c
Let Mn be an n−dimensional diferentiable manifold. Differantial transformation of algebra T (Mn ), defined by D = ∇X : T (Mn ) → T (Mn ), X ∈ =10 (Mn ), is called as covariant derivation with respect to vector field X if ∇f X+gY t ∇X f =00 (Mn ), ∀X, Y
= f ∇X t + g∇Y t, = Xf,
=10 (Mn ), ∀t
where ∀f, g ∈ ∈ ∈ =(Mn ). On the other hand, a transformation defined by ∇ : =10 (Mn ) × =10 (Mn ) → =10 (Mn ), is called as affin connection [5, 8]. We now assume that Mn is a manifold with an affine connection ∇. Then there exist a unique affine connection ∇c in =(Mn ) which satisfies ∇cX c Y c = (∇X Y )c
(1.19)
for any X, Y ∈ =10 (Mn ). This affine connection is called the complete lift of the affine connection ∇ to T (Mn ) and denoted by ∇c [8]. Proposition 1.1. For any X ∈ =10 (Mn ), f ∈ =00 (Mn ) and ∇c is the complete lift of the affine connection ∇ to T (Mn ) [8] i) ∇cX v f v = 0, ii) ∇cX v f c = (∇X f )v , iii) ∇cX c f v = (∇X f )v , ıv) ∇cX c f c = (∇X f )c . Proposition 1.2. For any X, Y ∈ =10 (Mn ) and ∇c is the complete lift of the affine connection ∇ to T (Mn ) [8] i) ∇cX v Y v = 0, ii) ∇cX v Y c = (∇X Y )v , iii) ∇cX c Y v = (∇X Y )v , ıv) ∇cX c Y c = (∇X Y )c . 2. Main Results Let an n−dimensional differentiable manifold Mn be endowed with a tensor field ϕ of type (1, 1), a vector field ξ, a 1−form η, I the identity and let them satisfy (2.1)
ϕ2 = −I + η ⊗ ξ,
ϕ(ξ) = 0,
ηoϕ = 0,
η(ξ) = 1.
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Then (ϕ, ξ, η) define almost contact structure on Mn [3, 6, 8]. From (2.1), we get on taking complete and vertical lifts (ϕc )2
= −I + η v ⊗ ξ c + η c ⊗ ξ v ,
ϕc ξ v
=
0, ϕc ξ c = 0, η v oϕc = 0,
c
c
=
0, η v (ξ v ) = 0, η v (ξ c ) = 1,
η c (ξ v )
=
1, η c (ξ c ) = 0.
(2.2)
η oϕ
We now define a (1, 1) tensor field J on T (Mn ) by J = ϕc − ξ v ⊗ η v + ξ c ⊗ η c .
(2.3)
Then it is easy to show that J 2 X v = −X v and J 2 X c = −X c , which give that J is an almost contact structure on T (Mn ). We get from (2.3) JX v
=
(ϕX)v + (η(X))v ξ c ,
JX c
=
(ϕX)c − (η(X))v ξ v + (η(X))c ξ c
for any X ∈ =10 (Mn ) [3]. Theorem 2.1. For ∇X the operator covarient derivation with respect to X, J ∈ =11 (T (Mn )) defined by (2.3) and η(Y ) = 0, we have i) (∇cX v J)Y v
=
0,
(∇cX v J)Y c (∇cX c J)Y v (∇cX c J)Y c
=
((∇X ϕ)Y )v + ((∇X η)Y )v ξ c ,
=
((∇X ϕ)Y )v + ((∇X η)Y )v ξ c ,
=
((∇X ϕ)Y )c − ((∇X η)Y )v ξ v + ((∇X η)Y )c ξ c ,
ii) iii) ıv)
where X, Y ∈ =10 (Mn ), a tensor field ϕ ∈ =11 (Mn ), a vector field ξ and a 1−form η ∈ =01 (Mn ). Proof. For J = ϕc − ξ v ⊗ η v + ξ c ⊗ η c and η(Y ) = 0, we get i) (∇cX v J)Y v
= ∇cX v (ϕc − ξ v ⊗ η v + ξ c ⊗ η c )Y v − (ϕc − ξ v ⊗ η v + ξ c ⊗ η c )∇cX v Y v = ∇cX v (ϕY )v − ∇cX v (η v (Y )v )ξ v + ∇cX v (η(Y ))v ξ c =
ii) (∇cX v J)Y c
0,
= ∇cX v (ϕc − ξ v ⊗ η v + ξ c ⊗ η c )Y c − (ϕc − ξ v ⊗ η v + ξ c ⊗ η c )∇cX v Y c = ∇cX v ϕc Y c − ∇cX v (ηY )v ξ v + ∇cX v (η(Y ))c ξ c − ϕc ∇cX v Y c +η v (∇X Y )v ξ v − (η(∇X Y ))v ξ c =
(∇cX v ϕc )Y c + ϕc (∇cX v Y c ) − ϕc ∇cX v Y c − (∇X (η(Y )))v ξ c +((∇X η)Y )v ξ c
= iii) (∇cX c J)Y v
(∇X ϕ)Y )v + ((∇X η)Y )v ξ c ,
= ∇cX c (ϕc − ξ v ⊗ η v + ξ c ⊗ η c )Y v − (ϕc − ξ v ⊗ η v + ξ c ⊗ η c )∇cX c Y v = ∇cX c ϕc Y v − ∇cX c (η v (Y )v )ξ v + ∇cX c (η(Y ))v ξ c − ϕc ∇cX c Y v +η v (∇X Y )v ξ v − (η(∇X Y ))v ξ c =
(∇cX c ϕc )Y v + ϕc (∇cX c Y v ) − ϕc ∇cX c Y v − (∇X (η(Y )))v ξ c +(∇X η)Y )v ξ c
=
(∇X ϕ)Y )v + ((∇X η)Y )v ξ c ,
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ıv) (∇cX c J)Y c
=
∇cX c (ϕc − ξ v ⊗ η v + ξ c ⊗ η c )Y c − (ϕc − ξ v ⊗ η v + ξ c ⊗ η c )∇cX c Y c
=
∇cX c ϕc Y c − ∇cX c ((ηY )v )ξ v + ∇cX c (η(Y ))c ξ c − ϕc ∇cX c Y c +(η(∇X Y ))v ξ v − (η(∇X Y ))c ξ c
=
(∇cX c ϕc )Y c + ϕc (∇cX c Y c ) − ϕc ∇cX c Y c + (∇X (η(Y )))v ξ v −((∇X η)Y )v ξ v − (∇X (η(Y )))c ξ c + ((∇X η)Y )c ξ c
=
(∇X ϕ)Y )c − ((∇X η)Y )v ξ v + ((∇X η)Y )c ξ c .
Corollary 2.1. If we put Y = ξ, i.e. η(ξ) = 1 and ξ has the conditions of (2.1), then we get different results i) (∇cX v J)ξ v
=
(∇X ξ)v ,
ii) (∇cX v J)ξ c
=
((∇X ϕ)ξ)v + (((∇X η))ξ)v ξ c ,
iii) (∇cX c J)ξ v
=
((∇X ϕ)ξ)v + (∇X ξ)c + ((∇X η)ξ)v ξ c ,
ıv) (∇cX c J)ξ c
=
(∇X ϕ)ξ)c − (∇X ξ)v − ((∇X η)ξ)v ξ v + ((∇X η)ξ)c ξ c .
Let an n−dimensional differentiable manifold Mn be endowed with a tensor field ϕ of type (1, 1), a vector field ξ, a 1−form η, I the identity and let them satisfy (2.4)
ϕ2 = I − η ⊗ ξ,
ϕ(ξ) = 0,
ηoϕ = 0,
η(ξ) = 1.
Then (ϕ, ξ, η) define almost paracontact structure on Mn [3, 6]. From (2.4), we get on taking complete and vertical lifts (ϕc )2
= I − ηv ⊗ ξc − ηc ⊗ ξv ,
ϕc ξ v
=
0, ϕc ξ c = 0, η v oϕc = 0,
η c oϕc
=
0, η v (ξ v ) = 0, η v (ξ c ) = 1,
=
1, η c (ξ c ) = 0.
(2.5)
c
v
η (ξ )
We now define a (1, 1) tensor field Je on T (Mn ) by Je = ϕc − ξ v ⊗ η v − ξ c ⊗ η c .
(2.6)
Then it is easy to show that Je2 X v = X v and Je2 X c = X c , which give that Je is an almost product structure on T (Mn ). We get from (2.6) e v JX e c JX
=
(ϕX)v − (η(X))v ξ c ,
=
(ϕX)v − (η(X))v ξ v − (η(X))c ξ c
for any X ∈ =10 (Mn ). Theorem 2.2. For ∇X the operator covarient derivation with respect to X, Je ∈ =11 (T (Mn )) defined by (2.6) and η(Y ) = 0, we have e v i) (∇cX v J)Y e c ii) (∇cX v J)Y
=
0,
=
((∇X ϕ)Y )v − ((∇X η)Y )v ξ c ,
e v iii) (∇cX c J)Y e c ıv) (∇c c J)Y
=
((∇X ϕ)Y )v − ((∇X η)Y )v ξ c ,
=
((∇X ϕ)Y )c − ((∇X η)Y )v ξ v − ((∇X η)Y )c ξ c ,
X
where X, Y ∈ =10 (Mn ), a tensor field ϕ ∈ =11 (Mn ), a vector field ξ ∈ =10 (Mn ) and a 1−form η ∈ =01 (Mn ).
COVARIENT DERIVATIVES OF ALMOST CONTACT STRUCTURE...
215
Proof. For Je = ϕc − ξ v ⊗ η v − ξ c ⊗ η c and η(Y ) = 0, we get e v i) (∇cX v J)Y
= ∇cX v (ϕc − ξ v ⊗ η v − ξ c ⊗ η c )Y v − (ϕc − ξ v ⊗ η v − ξ c ⊗ η c )∇cX v Y v = ∇cX v (ϕY )v − ∇cX v (η v (Y )v )ξ v − ∇cX v (η(Y ))v ξ c =
e c ii) (∇cX v J)Y
0,
= ∇cX v (ϕc − ξ v ⊗ η v − ξ c ⊗ η c )Y c − (ϕc − ξ v ⊗ η v − ξ c ⊗ η c )∇cX v Y c = ∇cX v ϕc Y c − ∇cX v (ηY )v ξ v − ∇cX v (η(Y ))c ξ c − ϕc ∇cX v Y c +η v (∇X Y )v ξ v + (η(∇X Y ))v ξ c =
(∇cX v ϕc )Y c + ϕc (∇cX v Y c ) − ϕc ∇cX v Y c + (∇X (η(Y )))v ξ c −((∇X η)Y )v ξ c
= e v iii) (∇cX c J)Y
(∇X ϕ)Y )v − ((∇X η)Y )v ξ c ,
= ∇cX c (ϕc − ξ v ⊗ η v − ξ c ⊗ η c )Y v − (ϕc − ξ v ⊗ η v − ξ c ⊗ η c )∇cX c Y v = ∇cX c ϕc Y v − ∇cX c (η v (Y )v )ξ v − ∇cX c (η(Y ))v ξ c − ϕc ∇cX c Y v +η v (∇X Y )v ξ v + (η(∇X Y ))v ξ c =
(∇cX c ϕc )Y v + ϕc (∇cX c Y v ) − ϕc ∇cX c Y v + (∇X (η(Y )))v ξ c −(∇X η)Y )v ξ c
e c ıv) (∇cX c J)Y
=
(∇X ϕ)Y )v − ((∇X η)Y )v ξ c ,
=
∇cX c (ϕc − ξ v ⊗ η v − ξ c ⊗ η c )Y c − (ϕc − ξ v ⊗ η v − ξ c ⊗ η c )∇cX c Y c
=
∇cX c ϕc Y c − ∇cX c ((ηY )v )ξ v − ∇cX c (η(Y ))c ξ c − ϕc ∇cX c Y c +(η(∇X Y ))v ξ v + (η(∇X Y ))c ξ c
=
(∇cX c ϕc )Y c + ϕc (∇cX c Y c ) − ϕc ∇cX c Y c + (∇X (η(Y )))v ξ v −((∇X η)Y )v ξ v + (∇X (η(Y )))c ξ c − ((∇X η)Y )c ξ c
=
(∇X ϕ)Y )c − ((∇X η)Y )v ξ v − ((∇X η)Y )c ξ c .
Corollary 2.2. If we put Y = ξ, i.e. η(ξ) = 1 and ξ has the conditions of (2.4), then we have e v = −(∇X ξ)v , i) (∇c v J)ξ X
e c ii) (∇cX v J)ξ e v iii) (∇cX c J)ξ
=
((∇X ϕ)ξ)v − (((∇X η))ξ)v ξ c ,
=
((∇X ϕ)ξ)v − (∇X ξ)c − ((∇X η)ξ)v ξ c ,
e c ıv) (∇cX c J)ξ
=
(∇X ϕ)ξ)c − (∇X ξ)v − ((∇X η)ξ)v ξ v − ((∇X η)ξ)c ξ c . References
[1] Blair, D. E., Contact Manifolds in Riemannian Geometry, Lecture Notes in Math, 509, Springer Verlag, New York, 1976. [2] Das, Lovejoy S., Fiberings on almost r-contact manifolds, Publicationes Mathematicae, Debrecen, Hungary 43(1993), 161-167. [3] Omran, T., Sharffuddin, A. and Husain, S. I., Lift of Structures on Manifolds, Publications de 1’Instıtut Mathematıqe, Nouvelle serie, 360(1984), no. 50, 93 – 97. [4] Oproiu, V., Some remarkable structures and connexions, defined on the tangent bundle, Rendiconti di Matematica 3(1973), 6 VI.
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[5] Salimov, A. A., Tensor Operators and Their applications, Nova Science Publ., New York, 2013. [6] Salimov, A. A. and C ¸ ayır, H., Some Notes On Almost Paracontact Structures, Comptes Rendus de 1’Acedemie Bulgare Des Sciences, 66(2013), no. 3, 331-338. [7] Sasaki, S., On The Differantial Geometry of Tangent Boundles of Riemannian Manifolds, Tohoku Math. J., 10(1958), 338-358. [8] Yano, K. and Ishihara, S., Tangent and Cotangent Bundles, Marcel Dekker Inc, New York, 1973 Department of Mathematics, Faculty of Arts and Sciences,, Giresun University, 28100, Giresun, Turkey E-mail address:
[email protected]
