Study of K- Hsu Structure in Generalized Almost Contact Manifold 2017

International Journal of Emerging Trends in Science and Technology IC Value: 76.89 (Index Copernicus) Impact Factor: 4.219 DOI: https://dx.doi.org/10.18535/ijetst/v4i8.30

Study of K- Hsu Structure in Generalized Almost Contact Manifold

Authors

Sahadat Ali

, Geeta Verma

Department of Mathematics, SRMGPC, Lucknow- 227105 Corresponding Author: Geeta Verma

1. Introduction

Let M₁,M₂,....,M_k be k-Hsu-structure manifolds each of class C^ and of dimension n₁,n₂,....,n_k respectively. Suppose (M₁)m₁,(M₂)m₂,....,(M_k)m_k, be their tangent spaces at

, M m ...., , M m , M

m₁ _{1 2} _{2 k} _k then the product space (M₁)m₁(M₂)m₂....(M_k)m_k,contains vector fields of the form (X₁,X₂,....,X_k),where X₁(M₁)m₁,X₂(M₂)m₂,....,X_k(M_k)m_k. Vector addition and scalar multiplication on above product space are defined as follows:

(1.1) (X₁,X₂,....,X_k)(Y₁,Y₂,....,Y_k)(X₁Y₁,X₂Y₂,....,X_kY_k) (1.2) (X₁,X₂,....,X_k)(X₁,X₂,....,X_k),

where X_i,Y_i(M_i)m_i, i1,2,....,kand is a scalar.

Under these conditions the product space(M₁)m₁(M₂)m₂....(M_k)m_k forms a vector space.

A linear transformation F on the product space is defined as (1.3) F(X₁,X₂,....,X_k)(F₁X₁,F₂X₂,....,F_k X_k),

where F₁,F₂,....,F_kare linear transformations on (M₁)m₁,(M₂)m₂,....,(M_k)m_krespectively.

If f₁,f₂,....,f_k be C^functions over the spaces (M₁)m₁,(M₂)m₂,....,(M_k)m_k respectively, we define the Abstract: Cartesian product of two manifolds has been defined and studied by Pandey[2]. In this paper we have taken Cartesian product of k-Hsu-Structure manifolds, where k is some finite integer, and studied some properties of curvature and Ricci tensor of such a product manifold.

Key words & Phases: k-Hsu-Structure manifolds, generalized almost contact structure, KH-structure.

Cfunction f₁,f₂,....,f_kon the product space as

(1.4) (X₁,X₂,....,X_k)(f₁,f₂,....,f_k)(X₁f₁,X₂f₂,....,X_kf_k).

Let D₁,D₂,....,D_kbe the connections on the manifoldsM₁,M₂,....,M_krespectively. We define the operator D on the product space as

(1.5) ( )( ) ( ₂ ).

2 2 1 1 1 2

1 2

1,X ,....,Xk Y,Y ,....,Yk DXY,D X Y ,....,DkXkYk

X

D 

Then D satisfies all four properties of a connection and thus it is a connection on the product manifold.

2. Some Results

Definition: Let there be defined onV_n, a vector valued linear function F of class C such that n

r I

a

F²  ^r _n 0 

where r is an integer and a is real or imaginary number. Then F is called Hsu-structure and V_nis called the Hsu-structure manifold.

Theorem 2.1: The product manifold M₁M₂....M_kadmits a Hsu-structure if and only if the manifolds Mk

...., M

M₁, ₂, are Hsu-structure manifolds.

Proof: Suppose M₁,M₂,....,M_kare Hsu-structure manifolds. Thus there exist tensor fieldsF₁,F₂,....,F_k each of type (1, 1) on M₁,M₂,....,M_krespectively satisfying

(2.1) F²i(X_i)a^rX_i,_n i1,2,....,k

where a is any complex number, not equal to zero.

In view of equation (1.3) it follows that there exists a linear transformation F onM₁M₂....M_kssatisfying (2.2) F²i(X₁, X₂,....,X_k)(F₁²X₁, F₂²X₂,....,F_k²X_k)

a^r(X₁, X₂,....,X_k) Thus, the product manifold admits a Hsu-structure.

Let us define a Riemannian metric g on the product manifold M₁M₂....M_kas (2.3) a^rg((X₁, X₂,....,X_k),(Y₁,Y₂,....,Y_k))a^rg₁(X₁,Y₁)a^rg₂(X₂,Y₂)....a^rg_k(X_k,Y_k)

where g₁,g₂,....,g_kare the Riemannian metrics over the manifolds M₁M₂....M_krespectively.

If  , ,...., be vector fields and  , ,...., be 1-forms on the Hsu-structure manifolds

Mk

...., M

M₁, ₂, respectively, then a vector field and a 1-form on the product manifold M₁,M₂,....,M_k is defined.

We now prove the following results.

Theorem 2.2: The product manifold M₁M₂....M_kadmits generalized almost contact structure if and only if the manifolds M₁,M₂,....,M_kpossess the same structure.

Proof: Let M₁,M₂,....,M_kare generalized almost contact manifolds. Thus there exists tensor fields F_iof type (1, 1) vector fields _iand 1-form. _i,i1,2,....,ksatisfying

(2.4) F_i²(X_i)a^rX_i_i(X_i)_i For product manifoldM₁M₂....M_k.

) ....,

( ) ...., , ,

( _{1 2 1}² _{1 2}² ₂ ²

2

k k

k F X ,F X , F X

X X

X

F 

By the help of equation (2.4), takes the form

), ) ( )

( ) ( ( ) ...., (

) ...., , ,

( _{1 2 1 2 1 1 1 2 2 2 k}

2 X X X_k a^r X ,X , X_k   X  , X  ,....,η_k X_k  F

or

(2.5) F²(X)a^rX(X).

Hence the product manifold admits a generalized almost contact structure.

Theorem 2.3: The product manifold M₁M₂....M_kadmits a KH-structure if and only if the manifolds Mk

...., M

M₁, ₂, are KH-structure manifolds.

Proof: Suppose M₁,M₂,....,M_kare KH-structure manifolds. Thus )

( ) (

) ( )

( _{2 2}

2 2 1 1 1

1 F Y D F Y

D_X  _X

(2.6) .........

= ……….

) ( )

( _{k k}

Xk

k F Y

 D

0

As D is a connection on the product manifold, we have

(2.7) (D_{(X1,X2,....,Xk)}F)(Y₁,Y₂,....,Y_k)D_{(X1,X2,....,Xk)}{F(Y₁,Y₂,....,Y_k)}

)}

...., , , (

{D_{(X1,X2,....,Xk)} Y₁Y₂ Y_k

F

In view of equation (1.3) and equation (1.5), this takes the form

) ...., , , ( )

...., , , ( )

(D_{(X1,X2,....,Xk)}F Y₁ Y₂ Y_k D_{(X1,X2,....,Xk)} F₁Y₁ F₂Y₂ F_kY_k

( ₂ )

2 2 1 1

1XY,D X Y ,....,DkXkYk

D

F

)

( _{2 2}

2 2 1 1 1

1X FY,D X FY ,....,DkXkFkYk

 D



)

( ₂

2 2 2 1 1 1

1DXY,FD X Y ,....,FkDkXkYk

 F

) ( ) (

) ( ) (

) ( )

(( _{2 2}

2 2 1 1 1

1X F Y , D X F Y ,...., DkXkFk Yk

 D

0.

Thus, the product manifold is KH-structure manifold.

Theorem 2.4: The product manifold M₁M₂....M_kof Hsu-structure manifolds M₁,M₂,....,M_kis almost Tachibana if and only if the manifolds M₁,M₂,....,M_kare separately Tachibana manifolds.

Proof: Let a Hsu-structure manifolds M₁,M₂,....,M_kare almost Tachibana manifolds. Then (2.8) D F Y D F_i Y_i , i , ,....,k.

Yi i i i i

iX )( ) ( )( ) 0 1 2

(   

3. Curvature and Ricci Tensor

Let X (X₁, X₂,....,X_k) and Y(Y₁,Y₂,....,Y_k) be C^ vector fields on the product manifoldM₁M₂....M_k and F(f₁, f₂,....,f_k)be a C^function. Then

(3.1) [(X₁, X₂,....,X_k),(Y₁,Y₂,....,Y_k)](f₁, f₂,....,f_k)

) , , ( )}

, , ( ) , , ){(

, ,

(X₁ X₂ ....,X_k Y₁ Y₂ ....,Y_k f₁ f₂ ....,f_k  Y₁ Y₂ ....,Y_k



).

] , ( , ] , ( , ] ,

[(X₁ Y₁ f₁ X₂ Y₂ f₂ ...., X_k Y_k f_k



Suppose K_i(X_i,Y_i,Z_i), i1,2,....,k be the curvature tensors of the Hsu-structure manifolds Mk

...., M

M₁, ₂, respectively. If K(X,Y,Z) be the curvature tensor of the product manifold .

M ....

M

M₁ ₂  _k Then we have

(3.2) K(X,Y,Z)[K₁(X₁,Y₁,Z₁),K₂(X₂,Y₂,Z₂),....,K_k(X_k,Y_k, Z_k)].

If W (W₁,W₂,....,W_k)be a vector field on the product manifold, then (3.3) K(X,Y,Z,W)g(K(X,Y,Z,W)),

(3.4) K(X,Y,Z,W)K₁(X₁,Y₁,Z₁,W₁)K₂(X₂,Y₂,Z₂,W₂)....K_k(X_k,Y_k,Z_k,W_k) Thus, we have

Theorem 3.1: The product of manifold M₁M₂....M_k.is of constant curvature if and only if Hsu-structure manifolds M₁,M₂,....,M_kare separately of constant curvature.

Theorem 3.2: The Ricci tensor of the product manifold M₁M₂....M_k.is the sum of the Ricci tensor of the Hsu-structure manifolds M₁,M₂,....,M_k

Theorem 3.3: The product of manifold M₁M₂....M_k.is an Einstein space if and only if the Hsu-structure manifolds M₁,M₂,....,M_kare separately Einstein space.

Proof: Let the product manifold M₁M₂....M_k.be an Einstein space. thus (3.5) Ric(X,Y)Cg (X,Y),

where ,

n

 K

C K being the scalar curvature and n being the dimension of the product manifold. Then .

k ...., , , i Y , X Cg Y , X

Ric( _{i i}) _i ( _{i i}), 1 2

Therefore the manifolds M₁,M₂,....,M_kare also Einstein spaces.

References

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2. H. B. Pandey, Cartesian product of two manifolds, Indian Journ. Pure Appl. Math, 12(1), 1981.

3. J. Pant, Hypersurface immersed in a GF-structure manifold, Demonstration Mathematica, 19(3), 1986.

4. Ram Nivas and Mohd. Nazrul Islam Khan, On submanifold immersed in Hsu-quarternion manifold, The Nepali Mathematical Science Report, 21 (1 – 2), 2003.

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