Integrability Conditions of a Generalised almost ... - Ultrascientist.org

322 Lata Bisht, et al.
Comparing equations (2.12) and (2.13), we get
the equations (2.11).
Corollary (2.2): We have
lm ml ln nl mn nm 0 , (2.14)
and
l 2 l , m
2 m , 2
n n . (2.15)
Proof: From equations (2.5)b and (2.11)b,
we get
x
lm q lQ 0 .
From equations (2.5)a and (2.11)a , we have
l
2
p
x
lP
x
x
p
x
P
x
l .
Similarly we can prove other relations.
Lemma (2.1): ( dl )( nX , nY ) 0 , (2.16)a
( dp x )( nX , nY ) 0 . (2.16)b
Proof: (2.11)c yields (2.16)a . For
( dl)( ( X ) , ( Y ) ) 0 .
Since d is skew symmetric in both the slots.
Proof of (2.16)b follows the same pattern.
1 1
,
a a
Lemma (2.2): 2(
dm)(
lX , lY ) [ lX . lY ] [ lX , lY ]
2
(2.16)a
1 1
( dm)(
lX , lY)
[ X , Y ] N(
X , Y )
6
a a
8
5
1
1 1
N ( X , Y ) [ X , Y ] [ X , Y ]
4 4
3
a
a a
1
+ [ X , Y ]
5 . (2.16)b
a
Proof: From equation (2.14) , we have
2(
dm)(
lX , lY ) 2m[
lX , lY ],
1 1
[ lX , lY ] [ lX , lY ]
2 .
a a
Now
1
1
8(
dm)(
lX , lY ) [2lX
,2lY
] [2lX
,2lY
]
2 ,
a
a
1 1 1 1 1
{ [ X , Y ] [ X , Y ] [ X , Y ] [ X , Y]}
2 4 3 3 2
a a a a a
1 1 1 1 1
+ { [ X , Y ] [ X , Y ] [ X , Y ] [ X , Y ]}
4 3
3
2 ,
a a a a a
1 1 1 1 1
[ X , Y ] [ X , Y ] [ X , Y ] [ X , Y ] [ X , Y ]
6 5 5 4 5
a a a a a
1 1 1
+ [ X , Y ] [ X , Y ] [ X , Y ]
4 4
3
,
a a a
1 1 1
[ X , Y ] {[ X , Y ] [ X , Y ] [ X , Y ] [ X , Y ]} [ X , Y ]
6 5 5
a a a