On The Structure Equation 2 0 p F F 

International Journal of Engineering Science Invention
ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726
www.ijesi.org ||Volume 5 Issue 9|| September 2016 || PP. 25-28
www.ijesi.org 25 | Page
On The Structure Equation
2
0p
F F 
1
Lakhan Singh, 2
Sunder Pal Singh
1
Department of Mathematics, D.J. College, Baraut, Baghpat (U.P.)
2
Department of Physics, D.J. College, Baraut, Baghpat (U.P.)
Abstract: In this paper, we have studied various properties of the F- sturcture manifold satisfying
2
0p
F F  where p is odd prime. Metric F-structure, kernel, tangent and normal vectors have also been
discussed.
Keywords: Differnetiable manifold, complementary projection operators, metric, kernel, tangent and normal
vectors.
I. Introduction
Let Mn
be a differentiable manifold of class C
and F be a (1,1) tensor of class C
, satisfying
(1.1)
2
0p
F F 
we define the operators l and m on Mn
by
(1.2)
2 2
1
1,p p
l F m I F 
    
where I is the identity operator.
From (1.1) and (1.2), we have
(1.3)
2 2
, , , 0l m I l l m m lm ml     
, 0,Fl lF F Fm mF   
Theorem (1.1): Let us define the (1,1) tensors p and q by
(1.4)
   2 2
1 /2 1 /2
, ,
p p
p m F q m F
 
   
Then p and q are invertible operators satisfying
(1.5)
3 3 2 2 2
, , , , 0pq I p q q p p q p p q I       
Proof: Using (1.2), (1.3) and (1.4)
(1.6)
2 2
,pq I p q m l    etc, the other results follow similarly
Theorem (1.2): Let us define the (1,1) tensors  and  by
(1.7)
2 2
1 1
, ,p p
m F m F  
    then
(1.8)
2
I  
Proof: Using (1.2), (1.3) and (1.7), we get the results
Theorem (1.3): Let us define the (1,1) tensors  and  by
(1.9)
2 2
1 1
, ,p p
l F l F  
    then
(1.10) 2 , 0n n
l  
Proof: Using (1.2), (1.3) and (1.9), 0, 
2 2
2 , 4 2 .... 2n n
l l l l     
II. Metric F-Structure
For F satisfying (1.1) and Riemannian metric g, let
(2.1)    , ,`F X Y g FX Y is skew symmetric then

2
0p
F F 
(2.2)    , ,g FX Y g X FY 
and  ,F g is called metric F-structure
Theorem (2.1): For F satisfying (1.1)
(2.3)
   
   
2 2
1 / 2 1 / 2
, , ,`
p p
g F X F Y g X Y m X Y
      
 
where
(2.4)      , , ,`m X Y g mX Y g X mY 
Proof: Using (1.2), (2.2), (2.4) and  2
1 / 2p  being a multiple of 4, we have
(2.5)
   
  
 
2 2 2 21 / 2 1 / 2 1 / 2 1
, 1 ,
p p p p
g F X F Y g X F Y
       
 
 ,g X lY 
 ,g X lY 
  ,g X I m Y  
   , ,`g X Y m X Y  
III. Kernel, Tangent And Normal Vectors
We define
(3.1)  : 0Ker F X FX 
(3.2)  :Tan F X FX X 
(3.3)   : , 0,Nor F X g X FY Y  
Theorem (3.1): For F satisfying (1.1) we have
(3.4)
2
2
... p
Ker F Ker F Ker F  
(3.5)
2
2
... p
Tan F Tan F Tan F  
(3.6)
2
2
... p
Nor F Nor F Nor F  
Proof: to (3.4) Using (1.1), let X Ker F
0FX 
2
0F X 
2
X Ker F 
Thus
(3.7)
2
Ker F Ker F
Now let
2 2
0X Ker F F X  
3
0F X 
………………………………………………
………………………………………………
2
0p
F X 
0FX 

2
0p
F F 
X Ker F 
Thus
(3.8)
2
Ker F Ker F
From (3.7) and (3.8) we get
(3.9)
2
Ker F Ker F
proceeding similarly we get (3.4)
Proof to (3.5): Let
X Tan F FX X  
 2 2
F X F X X   
2
X Tan F 
Thus
(3.10)
2
Tan F Tan F
Now let
2 2 2
X Tan F F X X  
3 3
F X X 
………………………………………………
………………………………………………
2
p p
F X X 
p
FX X  
p
FX X  
X Tan F 
Thus
(3.11)
2
Tan F Tan F
From (3.10) and (3.11)
(3.12)
2
Tan F Tan F proceeding similarly we get (3.5)
Proof to (3.6): Let
 , 0X Nor F g X FY  
 2
, 0g X F Y 
2
X Nor F 
Thus
(3.13)
2
Nor F Nor F
Now let
 2 2
, 0X Nor F g X F Y  
………………………………………………
………………………………………………
 2
, 0p
g X F Y 
 , 0g X FY 
X Nor F 
Thus

2
0p
F F 
(3.14)
2
Nor F Nor F
From (3.13) and (3.14), we get
(3.15)
2
,Nor F Nor F
Proceeding similarly we get (3.6)
References
[1]. A Bejancu: On semi-invariant submanifolds of an almost contact metric manifold. An Stiint Univ., "A.I.I. Cuza" Lasi Sec. Ia Mat.
(Supplement) 1981, 17-21.
[2]. B. Prasad: Semi-invariant submanifolds of a Lorentzian Para-sasakian manifold, Bull Malaysian Math. Soc. (Second Series) 21
(1988), 21-26.
[3]. F. Careres: Linear invairant of Riemannian product manifold, Math Proc. Cambridge Phil. Soc. 91 (1982), 99-106.
[4]. Endo Hiroshi : On invariant submanifolds of connect metric manifolds, Indian J. Pure Appl. Math 22 (6) (June-1991), 449-453.
[5]. H.B. Pandey & A. Kumar: Anti-invariant submanifold of almost para contact manifold. Prog. of Maths Volume 21(1): 1987.
[6]. K. Yano: On a structure defined by a tensor field f of the type (1,1) satisfying f3
+f=0. Tensor N.S., 14 (1963), 99-109.
[7]. R. Nivas & S. Yadav: On CR-structures and  2 3,2F   - HSU - structure satisfying
2 3 2
0r
F F

  ,
Acta Ciencia Indica, Vol. XXXVII M, No. 4, 645 (2012).
[8]. Abhisek Singh,: On horizontal and complete lifts Ramesh Kumar Pandey of (1,1) tensor fields F satisfying & Sachin Khare the
structure equation F  2 ,k S S =0. International Journal of Mathematics and soft computing. Vol. 6, No. 1 (2016), 143-152,
ISSN 2249-3328

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