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

324 Lata Bisht, et al.
1 1
[ lX , lY ] [ lX , lY ], (2.18)a
2
a a
1
1
1 1 1 1
N(
X , Y ) N(
X , Y ) [ X , Y ] [ X , Y ] [ X , Y ] [ X , Y ]
5 4
3
5
6
4
, (2.18)b
a
a
a a a a
[ lX , lY ] 0, (2.18)c
1 1 1
1 1 1 1
N(
X , Y ) N(
X , Y ) N(
X , Y ) [ X , Y ] [ X , Y ] [ X , Y ] [ X , Y ] [ X , Y ]
3 2
3
2
2
a
a a
a a a a
1
1 1 1
N(
X , Y ) [ X , Y ] [ X , Y ] [ X , Y ]
2 4
3
. (2.18)d
a
a a a
Proof: In order to that m is completely integrable, it is necessary and sufficient that
m=0, n=0 be completely integrable, that is
dm ( X , Y ) 0 , (2.19)a
dn ( X , Y ) 0 . (2.19)b
Now from equation (2.8) we have
l I
n
So dm ( lX , lY ) 0, (2.20)a
From equation (2.19)a , we get
dn ( lX , lY ) 0 . (2.20)b
dm ( lX , lY ) 0,
8dm ( lX , lY ) 0 .
From lemma (2.2) , we have
and
1 1
[ lX , lY ] [ lX , lY ]
2 ,
a a