Nijenhuis Tensor in Para Almost F-3 Structure Manifold

Nijenhuis Tensor in Para Almost F-3 Structure Manifold

Savita Patni

Nanhi Pari Seemant Engineering Institute, Pithoragarh

Abstract: In this Paper, I have defined Nijenhuis tensor in Para Almost F-3 Structure Manifold and decompositions of Nijenhuis tensor have also been done in this manifold.

1. INTRODUCTION Let F , x 1,2,3 be three structures on Vn, such that

   x

   rank ((F )) r , everywhere, (1.1)

   x

   (F)2 F F F F , x not summed, (1.2) a

   x y xyz x z xy x

   F(F)2 F F F , y not summed. (1.2) b

   x y xyz z y xy y

   Then F is called Para almost F-3 structure.

   x

   Consequently,

   F 3 F 0 in Para almost F-3 structure manifold. (1.3)

   x x

2. NIJENHUIS TENSOR The Nijenhuis tensor with respect to F is defined as

x

N ( X ,Y ) [F X , F Y ] F 2 [ X ,Y ] F[F X ,Y ] F[ X , F Y ] . (2.1)

x x x x x x x x

Theorem (2.1). In Para almost F-3 structure manifold, we have

N(F 2 X ,Y ) [F X , F Y ] F 2 [F 2 X ,Y ] F[F X ,Y ] F[F 2 X , F Y ], (2.2) a

1 1 1 1 1 1 1 1 1 1 1

N( X , F 2 Y ) [F X , F Y ] F 2 [ X , F 2 Y ] F[F X , F 2 Y ] F[ X , F Y ], (2.2) b

1 1 1 1 1 1 1 1 1 1 1

N(F 2 X , F 2 Y ) [F X , F Y ] F 2 [F 2 X , F 2 Y ] F[F X , F 2 Y ]

1 1 1

1 1 1 1 1

1 1 1

F[F 2 X , F Y ] , (2.2)c

1 1 1

N(F 2 X ,Y ) [F F X , F Y ] F 2 [F 2 X ,Y ] F[F F X ,Y ] F[ X , F F Y ], (2.2) d

1 2 3 2 1 1 2

1 3 2

1 3 2

N( X , F 2 Y ) [F X , F F Y ] F 2 [ X , F 2 Y ] F[F X , F 2 Y ] F[ X , F F Y ], (2.2)e

1 2 1 3 2 1 2

1 1 2

1 3 2

N(F 2 X , F 2 Y ) [F F X , F F Y ] F 2 [F 2 X , F 2 Y ] F[F F X , F 2 Y ]

1 2 2

3 2 3 2

1 2 2

1 3 2 2

F[F 2 X , F F Y ] , (2.2) f

1 2 3 2

N (F 2 X ,Y ) [F F X , F Y ] F 2[F 2 X ,Y ] F[F F X ,Y ] F[F 2 X , F Y ], (2.2) g

2 1 3 1 2 2 1

2 3 1

2 1 2

N (F 2 X , F 2 Y ) [F F X , F F Y ] F 2[F 2 X , F 2 Y ] F[F F X , F 2 Y ]

2 1 1

3 1 3 1

2 1 1

2 3 1 1

F[F 2 X , F F Y ] , (2.2) h

2 1 3 1

N ( X , F 2 Y ) [F X , F F 2 Y ] F 2[ X , F 2 Y ] F[F X , F 2 Y ]

2 3 2 2 3 2 3 2 2 3

F[ X , F F 2 Y ], (2.2) i

2 2 3

N (F 2 X , F 2 Y ) [F F X , F Y ] F 2[F 2 X , F 2 Y ] F[F F X , F 2 Y ]

2 3 2

1 3 2

2 3 2

2 1 3 2

F[F 2 X , F Y ], (2.2) j

2 3 2

N (F 2 X ,Y ) [F F X , F Y ] F 2[F 2 X ,Y ] F[F F X ,Y ] F[F 2 X , F Y ], (2.2) k

3 1 2 1 3 3 1

3 2 1

3 1 3

N (F 2 X , F 2 Y ) [F F X , F Y ] F 2[F 2 X , F 2 Y ] F[F F X , F 2 Y ]

3 2 3

1 2 3

3 2 3

3 1 2 3

F[F 2 X , F Y ]. (2.2) l

3 2 3

Proof. Operating

F 2 on X (or Y or X and Y together) in equation (2.1) and using the equation (1.3) in the resulting equations

1

then we obtain the equations (2.2)a, (2.2)b and (2.2)c. Operating F 2

2

(1.2)b, we get

on X for x 1 in equation (2.1), then using the equation

N (F 2 X ,Y ) [F F 2 X , F Y ] F 2[F 2 X ,Y ] F[F F 2 X ,Y ] F[F 2 X , F Y ]

1 2 1 2 1 1 2

1 1 2

1 2 1

[F F X , F Y ] F 2[F 2 X ,Y ] F[F F X ,Y ] F[F 2 X , F Y ] ,

3 2 1 1 2

1 3 2

1 2 1

which is the equation (2.2)d. Similarly, we can obtain the equations (2.2)e and (2.2)f.

Now, operating

F 2 on X for

1

x 2 in equation (2.1) and using the equation (1.2) b, we get (2.2) g. Proof of the equations

(2.2) h,.., (2.2) l follows similarly.

Theorem (2.2). If we put

P( X ,Y ) [F X , F Y ] F[F X ,Y ]. (2.3)

x

Then

x x x x

P(F 2 X ,Y ) [F X , F Y ] F[F X ,Y ] , (2.4) a

1 1 1 1 1 1

P( X , F 2 Y ) [F X , F Y ] F[F X , F 2 Y ] , (2.4) b

1 1 1 1 1 1 1

P(F 2 X , F 2 Y ) [F X , F Y ] F[F X , F 2 Y ] , (2.4) c

1 1 1 1 1 1 1 1

P(F 2 X ,Y ) [F F X , F Y ] F[F F X ,Y ] , (2.4) d

1 2 3 2 1 1 3 2

P( X , F 2 Y ) [F X , F F Y ] F[F X , F 2 Y ] , (2.4) e

1 2 1 3 2 1 1 2

P(F 2 X , F 2 Y ) [F F X , F F Y ] F[F F X , F 2 Y ] , (2.4) f

1 2 2

3 2 3 2

1 3 2 2

P(F 2 X , F 2 Y ) [F X , F F Y ] F[F X , F 2 Y ], (2.4) g

1 1 2

1 3 2

1 1 2

F P(F 2 X , F 2 Y ) F[F X , F F Y ] F 2 [F X , F 2 Y ], (2.4) h

1 1 1 2

1 1 3 2

1 1 2

F P(F X , F 2 Y ) F[F 2 X , F F Y ] F 2 [F 2 X , F 2 Y ], (2.4) i

1 1 1 2

1 1 3 2

1 1 2

P( X , F 2 Y ) [F X , F F Y ] F[F X , F 2 Y ] , (2.4) j

2 1 2 3 1 2 2 1

P(F 2 X , F 2 Y ) [F X , F F Y ] F[F X , F 2 Y ] , (2.4) k

2 2 1

2 3 1

2 2 1

P( X , F 2 Y ) [F X , F F Y ] F[F X , F 2 Y ] , (2.4) l

2 3 2 1 3 2 2 3

P(F 2 X , F 2 Y ) [F X , F F Y ] F[F X , F 2 Y ] , (2.4) m

2 2 3

2 1 3

2 2 3

P( X , F 2 Y ) [F X , F F Y ] F[F X , F 2 Y ] , (2.4) n

3 1 3 2 1 3 3 1

P(F 2 X , F 2 Y ) [F X , F F Y ] F[F X , F 2 Y ] , (2.4) o

3 3 1

3 2 1

3 3 1

P( X , F 2 Y ) [F X , F F Y ] F[F X , F 2 Y ] , (2.4) p

3 2 3 1 2 3 3 2

P(F 2 X , F 2 Y ) [F X , F F Y ] F[F X , F 2 Y ] . (2.4) q

3 3 2

Consequently

3 1 2

3 3 2

P( X ,Y ) P(F 2 X ,Y ) [F X , F Y ] F[F X ,Y ] , (2.5) a

1 1 1 1 1 1 1

P( X , F 2 Y ) P(F 2 X , F 2 Y ) [F X , F Y ] F[F X , F 2 Y ] , (2.5) b

1 1 1 1 1 1 1 1 1 1

P( X , F 2 Y ) P(F 2 X , F 2 Y ) [F X , F F Y ] F[F X , F 2 Y ], (2.5) c

1 2 1 1 2

1 3 2

1 1 2

P( X , F 2 Y ) P(F 2 X , F 2 Y ) [F X , F F Y ] F[F X , F 2 Y ] , (2.5) d

2 1 2 2 1

2 3 1

2 2 1

P( X , F 2 Y ) P(F 2 X , F 2 Y ) [F X , F F Y ] F[F X , F 2 Y ]

(2.5) e

2 3 2 2 3

2 1 3

2 2 3

P( X , F 2 Y ) P(F 2 X , F 2 Y ) [F X , F F Y ] F[F X , F 2 Y ] , (2.5) f

3 1 3 3 1

3 2 1

3 3 1

P( X , F 2 Y ) P(F 2 X , F 2 Y ) [F X , F F Y ] F[F X , F 2 Y ] . (2.5) g

3 2 3 3 2

3 1 2

3 3 2

Proof. Operating

F 2 on X (or Y or X and Y together) for

1

x 1 in equation (2.3) and using the equation (1.3), we get (2.4)a,

(2.4)b and (2.4)c. Similarly, operating

F 2 on X (or Y or X and Y together) for

2

x 1and using (1.2)b, we obtain the equations

(2.4)d, (2.4)e and (2.4)f. Equation (2.4)g is obtained by operating

F 2 on X and F 2

1 2

on Y for

x 1 then using the equation

(1.2)b in the resulting equation. Proof of the equations (2.4)h, , (2.4)o follows similarly.

Comparing the equations (2.4)a and (2.4)b, we get (2.5)a. Equation (2.5)b is obtained by comparing the equations (2.4)c and (2.4)d. Similarly, we can obtain the remaining equations.

Theorem (2.3). If we put

Q( X ,Y ) F 2 [ X ,Y ] F[ X , F Y ] . (2.6).

x x x x

Then

Q(F 2 X ,Y ) F 2 [F 2 X ,Y ] F[F 2 X , F Y ] , (2.7) a

1 1 1 1

1 1 1

Q(F 2 X , F 2 Y ) F 2 [F 2 X , F 2 Y ] F[F 2 X , F Y ] , (2.7) b

1 1 1

1 1 1

1 1 1

F 2 Q( X ,Y ) F 2 [ X ,Y ] F[ X , F Y ] , (2.7) c

1 1 1 1 1

F 2 Q( X , F 2 Y ) F 2 [ X , F 2 Y ] F[ X , F Y ] , (2.7) d

1 1 1

1 1 1 1

Q(F 2 X , F 2 Y ) F[F 2 X , F F Y ] F 2 [F 2 X , F 2 Y ], (2.7) e

1 1 2

1 1 3 2

1 1 2

Q(F 2 X , F 2 Y ) F 2 [F 2X , F 2 Y ] F[F 2 X , F F Y ], (2.7) f

1 2 2

1 2 2

1 2 3 2

Q(F 2 X , F 2 Y ) F 2[F 2 X , F 2 Y ] F[F 2 X , F F Y ] , (2.7) g

1 2 3

1 2 3

1 2 2 3

Q(F 2 X , F 2 Y ) F 2[F 2 X , F 2 Y ] F[F 2 X , F F Y ] , (2.7) h

2 2 1

2 2 1

2 2 3 1

Q(F 2 X , F 2 Y ) F 2[F 2 X , F 2 Y ] F[F 2 X , F F Y ] , (2.7) i

2 1 3

2 1 3

2 1 1 3

Q(F 2 X , F 2 Y ) F 2[F 2 X , F 2 Y ] F[F 2 X , F F Y ] , (2.7) j

3 3 1

3 3 1

3 3 2 1

Q(F 2 X , F 2 Y ) F 2[F 2 X , F 2 Y ] F[F 2 X , F F Y ] , (2.7) k

3 3 2

3 3 2

3 3 1 2

F 2 Q(F 2 X , F 2 Y ) F 2[F 2 X , F 2 Y ] F[F 2 X , F F Y ] , (2.7) l

1 1 2 3

1 2 3

1 2 2 3

F 2 Q(F 2 X , F 2 Y ) F 2[F 2 X , F 2 Y ] F[F 2 X , F F Y ] , (2.7) m

2 2 2 1

2 2 1

2 2 3 1

F 2 Q(F 2 X , F 2 Y ) F 2[F 2 X , F 2 Y ] F[F 2 X , F F Y ] , (2.7) n

2 2 1 3

2 1 3

2 1 1 3

F 2 Q(F 2 X , F 2 Y ) F 2[F 2 X , F 2 Y ] F[F 2 X , F F Y ] , (2.7) o

3 3 3 1

3 3 1

3 3 2 1

F 2 Q(F 2 X , F 2 Y ) F 2[F 2 X , F 2 Y ] F[F 2 X , F F Y ] . (2.7) p

3 3 3 2

Consequently

3 3 2

3 3 1 2

Q(F 2 X ,Y ) F 2 Q(F 2 X ,Y ) F 2 [ X , F 2 Y ] F[ X , F Y ] , (2.8) a

1 1 1 1 1

1 1 1 1

Q( X , F 2 Y ) F 2 Q( X , F 2 Y ) F 2 [ X , F 2 Y ] F[ X , F Y ] , (2.8) b

1 1 1 1 1

1 1 1 1

Q(F 2 X , F 2 Y ) F 2 Q(F 2 X , F 2 Y ) F 2 [F 2 X , F 2 Y ] F[F 2 X , F Y ], (2.8) c

1 1 1

1 1 1 1

1 1 1

1 1 1

Q(F 2 X , F 2 Y ) F 2 Q(F 2 X , F 2 Y ) F 2[F 2 X , F 2 Y ] F[F 2 X , F F Y ] , (2.8) d

1 2 3

1 1 2 3

1 2 3

1 2 2 3

Q(F 2 X , F 2 Y ) F 2 Q(F 2 X , F 2 Y ) F 2[F 2 X , F 2 Y ] F[F 2 X , F F Y ] , (2.8) e

2 2 1

2 2 2 1

2 2 1

2 2 3 1

Q(F 2 X , F 2 Y ) F 2 Q(F 2 X , F 2 Y ) F 2[F 2 X , F 2 Y ] F[F 2 X , F F Y ] , (2.8) f

2 1 3

2 2 1 3

2 1 3

2 1 1 3

Q(F 2 X , F 2 Y ) F 2 Q(F 2 X , F 2 Y ) F 2[F 2 X , F 2 Y ] F[F 2 X , F F Y ] , (2.8) g

3 3 1

3 3 3 1

3 3 1

3 3 2 1

Q(F 2 X , F 2 Y ) F 2 Q(F 2 X , F 2 Y ) F 2[F 2 X , F 2 Y ] F[F 2 X , F F Y ] . (2.8) h

3 3 2

3 3 3 2

3 3 2

3 3 1 2

Proof. The proof follows the pattern of the proof of the theorem (2.2).

Corollary (2.1). In the Para almost F-3 structure manifold, we have

Q(F X , F 2 Y ) F P(F 2 X , F 2 Y ) F[F X , F F Y ] F 2 [F X , F 2 Y ], (2.9) a

1 1 2

1 1 1 2

1 1 3 2

1 1 2

Q(F 2 X , F 2 Y ) F P(F X , F 2 Y ) F[F 2 X , F F Y ] F 2 [F 2 X , F 2 Y ] , (2.9) b

1 1 2

1 1 1 2

1 1 3 2

1 1 2

F 2 P(F X , F Y ) Q(F 2 X , F 2 Y ) F 2[F 2 X , F 2 Y ] F[F 2 X , F Y ] , (2.9) c

1 1 1 1

1 1 1

1 1 1

1 1 1

P(F 2 X ,Y ) Q(F 2 X ,Y ) N (F 2 X ,Y )

(2.9) d

1 1 1 1 1 1

P( X , F 2 Y ) Q( X , F 2 Y ) N ( X , F 2 Y ) , (2.9) e

1 1 1 1 1 1

P(F 2 X , F 2 Y ) Q(F 2 X , F 2 Y ) N (F 2 X , F 2 Y ) , (2.9) f

1 1 1

1 1 1

1 1 1

P(F 2 X ,Y ) Q(F 2 X ,Y ) N (F 2 X ,Y ) , (2.9) g

1 2 1 2 1 2

P(F 2 X , F 2 Y ) Q(F 2 X , F 2 Y ) N (F 2 X , F 2 Y ) . (2.9) h

1 2 2

1 2 2

1 2 2

Proof. Comparing the equations (2.4)h and (2.7)h, we get (2.9)a. Similarly comparing (2.4)i and (2.7)i, we get (2.9)b. Operating

F on X and Y in (2.3) for x 1, we get

1

P(F X , F Y ) [F 2 X , F 2 Y ] F[F 2 X , F Y ] . (2.10)

1 1 1 1 1 1 1 1

Operating

F 2 in (2.10) throughout and using (1.3) then comparing the resulting equation with (2.7)c, we obtain (2.9)c. Adding

1

the equations (2.4)a and (2.7)a then comparing the resulting equation with (2.2)g, we get (2.9)d. Proof of the remaining equations follows similarly.

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