On Integrablity Of F-Structure Satisfying F 2K+1 +F=0

The International Journal Of Engineering And Science (IJES) || Volume || 5 || Issue || 9 || Pages || PP 64-66 || 2016 || ISSN (e): 2319 – 1813 ISSN (p): 2319 – 1805

On Integrablity Of F-Structure Satisfying F2K+1+F=0 Lakhan Singh Department of Mathematics, D.J. College, Baraut, Baghpat (U.P.) -------------------------------------------------------- ABSTRACT------------------------------------------------------------The purpose of this paper is to study integrability of the F-structure satisying F2K+1 + F=0, where K is a positive integer. Nijenhuis tensor, metric F-structure, fundamental 2-form have also been discussed.

Key words: Differentiable manifold, projection operators, tangent bundle, metric and 2-form.

--------------------------------------------------------------------------------------------------------------------------------------Date of Submission: 17 May 2016 Date of Accepted: 22 August 20 16 ----------------------------------------------------------------------------------------------------------------------------- --------I. Let

Vn

be a



C 

INTRODUCTION

differentiable manifold and F be a

C  (1,1) tensor satisfying

F 2k 1  F  0,

1.1

we define the projection operators l and m by

l   F 2K , m  I  F 2K ,

1.2

From (1.1) and (1.2) we have

l  m  I , l 2  l , m2  m, lm  ml  0 Fl  lF  F , Fm  mF  0

(1.3)

II. Let

N l

(2.1)

and

N

NIJENHUIS TENSOR

denote the Nijenhuis tensors corresponding to the operators l and m respectively, then

m

N  X ,Y   lX , lY   l 2  X ,Y   l lX ,Y   l  X , lY  l

(2.2)

N  X ,Y    mX , mY   m2  X ,Y   mmX ,Y   m X , mY  . m

Theorem (2.1) For the F structure satisfying (1.1), we have (2.3)

N  lX , lY   m lX , lY  l

(2.4)

N  lX , mY   0 l

(2.5)

N  mX , lY   0 l

(2.6)

N  mX , mY   l  mX , mY  l

Proof: Using (1.3) and (2.3) (2.7)

N  lX , lY   l 2 X , l 2Y   l 2 lX , lY   l l 2 X , lY   l lX , l 2Y  l

 lX , lY   l lX , lY   l lX , lY   l lX , lY    I  l  lX , lY   m lX , lY 

Proceeding similarly we get other results.

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On Integrablity Of F-Structure Satisfying III.

INTEGRABILITY CONDITIONS

Theorem (3.1): The necessary aand sufficient condition that the tangent bundle (3.1)

N  lX , lY   0

l*

is integrable, is

l

l * is integrable if mX  0 for X  lX   dm X ,Y   0 for X  lX , Y  lY

Proof: (3.2)

(3.3)

  dmlX , lY   0, simplyfing it, we get mlX , lY   0

From (2.3) and (3.3), we get (3.1) Theorem (3.2): The necessary and sufficient condition that the tangent bundle (3.4)

N  mX , mY   0

m*

is integrable, is

l

m* is integrable if lX  0 for X  mX   dl  X ,Y   0, for X  mX , Y  mY

Proof: (3.5)

(3.6)

  dl  mX , mY   0 l  mX , mY   0

simplifying it, we get

From (2.6) and (3.6), we get (3.4) Theorem (3.): The necessary and sufficient condition that Vn regarded as the sum of is (3.7)

l*

and

m*

is integrable,

N  X ,Y   0 l

Proof: Using (1.3), we have (3.8)

N  X , Y   N  lX , lY   N  lX , mY   N  mX , lY  l

l

l

 N  mX , mY 

l

l

Using (2.4) and (2.5) in (3.8), (3.9)

N  X , Y   N  lX , lY   N  mX , mY  l

l

l

Using theorem (3.1), theorem (3.2) in (3.9) the necessary and sufficient condition for integrability of Vn is vanishing of

N  X ,Y  l

IV.

METRIC F-STRUCTURE

Let g be the Riemannian metric such that (4.1)

`F  X ,Y   g  FX ,Y  is skew symmetric then

(4.2)

g  FX ,Y    g  X , FY  also F , g is called a metric F-structure.

Theorem (4.1) For the structure F satisfying (1.1), we have (4.3)

 DX `F   F 2 K Y , Z   g  DX F 2 KY , Z   g  DX FY , Z 

Proof: Using (1.1), (4.1) and (4.2) we get (4.3)

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On Integrablity Of F-Structure Satisfying REFERENCE [1].

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Ram Nivas and Surendra Yadav (2012): On CR structure and Fλ(2V+3,2) –Hsu-stucture statisfying F2v+3+λrF2=0, Acta Ciencia Indica, Vol. XXXVIIIM, No. 4, 645.

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