In the Semi-Riemannian case we do not have the liability of the existence of such a metric being a difference from the Riemannian case. A Semi-Riemannian manifold provided with a normal contact metric structure is called Sasakian manifold.Semi-Riemannian, Sasakian manifold

Catalin Angelo Ioan

English

Research Papers in Economics

ŒCONOMICA 
 
  33
 
 
 
DEGENERATE FOLIATIONS IN SASAKIAN 
SEMI-RIEMANNIAN MANIFOLDS 
 
 
Associate Professor Cătălin Angelo IOAN, PhD 
“Danubius” University from Galati  
 
Abstract: In the Semi-Riemannian case we do not have the liability of the 
existence of such a metric being a difference from the Riemannian case. A Semi-
Riemannian  manifold  provided  with  a  normal  contact  metric  structure  is  called 
Sasakian manifold. 
Keywords: Semi-Riemannian, Sasakian manifold 
Jel  Classification:  C  -  Mathematical  and  Quantitative  Methods,  C0  - 
General‚ C00 - General  
 
 
In the Semi-Riemannian case we do not have the liability of the existence of 
such a metric being a difference from the Riemannian case. 
Definition  Let  M=M
2n+1  a  connected  Semi-Riemannian  manifold  and  j,  x,  h 
tensorial fields on M of types (1,1), (1,0) and (0,1) respectively. We say that (j,x,h) 
define an almost contact structure on M if: 
(1)        j
2Y=-Y+h(Y)x "YÎX (M) 
(2)              h(x)=1 
  In these conditions we have the relations: 
(3)             h(jY)=0 "YÎX  (M) 
(4)               jx=0 
  We define the contact distribution like D={YÎX (M)h(Y)=0}. 
Definition A metric g on M for which: 
(5)      g(jX,jY)=g(X,Y)-eh(X)h(Y) "X,YÎX  (M) 
where eÎ{-1,1} is called compatible metric with the almost contact structure. 
  In the Semi-Riemannian case we do not have the liability of the existence of 
such a metric being a difference from the Riemannian case. ACTA UNIVERSITATIS DANUBIUS                                                       Nr. 1/2005 
 
  34 
Definition  If  on  a  Semi-Riemannian  manifold  M
2n+1  we  have  an  almost  contact 
structure  (j,x,h)  and  a  compatible  metric  with  the  structure  we  shall  say  that 
(j,x,h,g) define an almost contact metric structure. 
  For Y=x in (1) we have with (2) and (4): 
(6)              g(X,x)=eh(X) "XÎX (M) 
and if in (6) Y=x: 
(7)            g(x,x)=e 
Remark If we shall consider in the upper g'=-g, x'=-x, h'=-h and j'=j we have 
together with e'=-e also an almost contact metric structure. We can further suppose 
that e=1. 
Definition:  Let M=M
2n+1 a connected Semi-Riemannian manifold provided with an 
almost contact metric structure (j,x,h,g). We shall say that the structure is contact 
metric if: 
(8)        (dh)(X,Y)=g(jX,Y) "X,YÎX  (M) 
Definition: An almost contact metric structure (j,x,h,g) is normal if: 
(9)                    (ÑXj)Y= g(X,Y)x-h(Y)X "X,YÎX  (M) 
where Ñ is the Levi-Civita connection for the metric g. 
Definition  A  Semi-Riemannian  manifold  provided with  a  normal  contact  metric 
structure is called Sasakian manifold. 
  On a Sasakian manifold the following relations hold: 
(10)          ÑXx=-jX "XÎX  (M) 
(11)      N
1(X,Y)=Nj(X,Y)+2dh(X,Y)x=0 "X,YÎX  (M) 
(12)         N
2(X,Y)=(LjXh)Y-(LjYh)X=0 "X,YÎX  (M) 
(13)        N
3(X)=(Lxj)X=0 "XÎX  (M) 
(14)        N
4(X)=(Lxh)X=0 "XÎX  (M) 
where Nj(X,Y)=[jX,jY]-j[jX,Y]-j[X,jY]+j
2[X,Y]. 
Let  now  a  j-basis  of TM  that is  an  orthonormal  basis of  the  form:  {x, 
Y1,...,Yn,jY1,...,jYn} where YiÎD, i=1,...,n. We denote by 
ei=g(Yi,Yi)=g(jYi,jYi), i=1,...,n. 
  We define now the 1-differential forms: 
(15)        wi(Z)=g(Yi,Z) "ZÎX  (M) "i=1,...,n 
and from (15) we have: 
(16)             g(jYi,Z)=-wi(jZ) "ZÎX  (M) "i=1,...,n 
  Let now two arbitrary vector fields: 
(17)      X=h(X)x+ ∑ ∑
= =
j j w e - w e
n
1 i
i i i
n
1 i
i i i Y ) X ( Y ) X (  ŒCONOMICA 
 
  35
(18)      Z=h(Z)x+ ∑ ∑
= =
j j w e - w e
n
1 j
j j j
n
1 j
j j j Y ) Z ( Y ) Z (  
We have now: 
(19) [X,Z]={h(X)Ñxh(Z)-h(Z)Ñxh(X)+∑
=
h Ñ w e
n
1 j
Y j j ) Z ( ) X (
j - 
∑
=
h Ñ w e
n
1 j
Y j j ) X ( ) Z (
j +∑
=
j h Ñ j w e
n
1 j
Y j j ) X ( ) Z (
j -∑
=
j h Ñ j w e
n
1 j
Y j j ) Z ( ) X (
j }x+ 
∑
=
e
n
1 j
j {h(X)Ñxwj(Z)-h(Z)Ñxwj(X)+∑
=
w Ñ w e
n
1 i
j Y i i ) Z ( ) X (
i -
∑
=
w Ñ w e
n
1 i
j Y i i ) X ( ) Z (
i +∑
=
j w Ñ j w e
n
1 i
j Y i i ) X ( ) Z (
i -
∑
=
j w Ñ j w e
n
1 i
j Y i i ) Z ( ) X (
i }Yj+∑
=
e
n
1 j
j {h(Z)Ñxwj(jZ)-h(X)Ñxwj(jZ)+∑
=
j w Ñ w e
n
1 i
j Y i i ) X ( ) Z (
i -
∑
=
j w Ñ w e
n
1 i
j Y i i ) Z ( ) X (
i + ∑
=
j j w Ñ j w e
n
1 i
j Y i i ) Z ( ) X (
i -∑
=
j j w Ñ j w e
n
1 i
j Y i i ) X ( ) Z (
i }jYj+ 
∑
=
e
n
1 j
j {h(X)wj(Z)-h(Z)wj(X)}[x,Yj]+∑
=
e
n
1 j
j {h(Z)wj(jX)-h(X)wj(jZ)}[x,jYj]-
∑
=
e e
n
1 j , i
j i wi(Z)wj(X)[Yi,Yj]+∑
=
e e
n
1 j , i
j i {wi(Z)wj(jX)-wi(X)wj(jZ)}[Yi,jYj]-
∑
=
e e
n
1 j , i
j i wi(jZ)wj(jX)[jYi,jYj]. 
  If we define now: 
(20)        Zj=wj(Z)X-wj(X)Z "j=1,...,n 
(21)             Wj=wj(jZ)X-wj(jX)Z "j=1,...,n 
(22)                T=h(X)Z-h(Z)X 
we obtain: 
(23) [X,Z]=(div Z)X-(div X)Z-{div T-2∑
=
w e
n
1 j
j j j ) W ( }x+∑
=
e
n
1 j
j j j Y )  Z div ( - 
∑
=
j e
n
1 j
j j j Y )  W div ( +∑
=
h e
n
1 j
j j ) Z ( [x,Yj]-∑
=
h e
n
1 j
j j ) W ( j[x,Yj]-∑
=
e e
n
1 j , i
j i wj(Wi)[jYi,Yj]- 
∑
<
=
e e
n
j i
1 j , i
j i wj(jWi)j[jYi,Yj]+∑
>
=
e e
n
j i
1 j , i
j i wi(jWj)j[jYi,Yj]+∑
<
=
e e
n
j i
1 j , i
j i {wi(Zj)-wj(jWi)}[Yi,Yj]. 
  In order that the two vector fields define a foliation we must have: 
(24) -{div T-2∑
=
w e
n
1 j
j j j ) W ( }x+∑
=
e
n
1 j
j j j Y )  Z div ( -∑
=
j e
n
1 j
j j j Y )  W div ( + ACTA UNIVERSITATIS DANUBIUS                                                       Nr. 1/2005 
 
  36 
∑
=
h e
n
1 j
j j ) Z ( [x,Yj]-∑
=
h e
n
1 j
j j ) W ( j[x,Yj]-∑
=
e e
n
1 j , i
j i wj(Wi)[jYi,Yj]-∑
<
=
e e
n
j i
1 j , i
j i wj(jWi)j[jYi,Yj]+ 
∑
>
=
e e
n
j i
1 j , i
j i wi(jWj)j[jYi,Yj]+∑
<
=
e e
n
j i
1 j , i
j i {wi(Zj)-wj(jWi)}[Yi,Yj]ÎSpan(X,Z). 
Example 
  Let  on  Rn
2n+1  with  the  coordinates  (x
1,...,x
n,y
1,...,y
n,z)  the  usual  contact 
structure defined by: 
(25)          h=dz-∑
=
n
1 i
i idx y  
(26)                x=
z ¶
¶  
(27)              j=










-
0 y 0
0 0 I
0 I 0
t
 
where y
t=(y
1,...,y
n) and I the identity. 
  The contact distribution D is generated by 
n ,..., 1 i
i
i i z
y
x
,
y
= 





¶
¶
+
¶
¶
¶
¶ . 
  The compatible metric is: 
(28)    g= ( )
2
n
1 i
2 i
i
n
1 i
i i
n
1 j , i
j i j i
ij i dz dy dz dx y 2 dx dx ) y y ( + e + - + d e ∑ ∑ ∑
= = =
 
Let 
(29)        X=
z
c
y
b
x
a
n
1 i
i i
n
1 i
i i ¶
¶
+
¶
¶
+
¶
¶
∑ ∑
= =
 
(30)        Z=
z
f
y
e
x
d
n
1 i
i i
n
1 i
i i ¶
¶
+
¶
¶
+
¶
¶
∑ ∑
= =
 
where ai,bi,c,di,ei,fÎF  (M). 
  We have: wi=eidy
i "i=1,...,n. It is easy to see that: 
(6.31)  wi(X)=eibi, wi(jX)=-eiai, wi(Z)=eiei, wi(jZ)=-eidi, h(X)=c-∑
=
n
1 i
i
iy a , 
h(Z)=f-∑
=
n
1 i
i
iy d  
  From (20)-(22) we have: 
(32)         Zj= 





¶
¶
- +
¶
¶
- +
¶
¶
- e ∑ ∑
= = z
) fb ce (
y
) b e e b (
x
) b d e a ( j j
n
1 i
i j i j i
n
1 i
i j i j i j  ŒCONOMICA 
 
  37
(33)         Wj= 





¶
¶
- +
¶
¶
- +
¶
¶
- e ∑ ∑
= = z
) cd fa (
y
) d b a e (
x
) d a a d ( j j
n
1 i
i j i j i
n
1 i
i j i j i j  
(34)   
z
) fa cd (
y
y ) e a d b ( ) f b c e (
x
y ) d a d a ( ) f a c d ( T
n
1 i
i i
n
1 i
i
n
1 j
j
i j j i i i
n
1 i
i
n
1 j
j
i j j i i i
¶
¶
-
+
¶
¶






- + - +
¶
¶






- + - =
∑
∑ ∑ ∑ ∑
=
= = = =  
  For a field U=
z
U
y
U
x
U
1 n 2
n
1 i
i
i n
n
1 i
i
i
¶
¶
+
¶
¶
+
¶
¶ +
=
+
= ∑ ∑  we have 
div U=
z
U
y
U
x
U
1 n 2 n
1 i
i
i n n
1 i
i
i
¶
¶
+
¶
¶
+
¶
¶
+
=
+
= ∑ ∑  
  therefore: 
(35) div Zj= 





¶
- ¶
+
¶
- ¶
+
¶
- ¶
e ∑ ∑
= = z
) fb ce (
y
) b e e b (
x
) b d e a ( j j
n
1 i
i
j i j i
n
1 i
i
j i j i
j  
(36) div Wj= 





¶
- ¶
+
¶
- ¶
+
¶
- ¶
e ∑ ∑
= = z
) cd fa (
y
) d b a e (
x
) d a a d ( j j
n
1 i
i
j i j i
n
1 i
i
j i j i
j  
∑ ∑ ∑
∑ ∑
= = =
= =
¶
- ¶
+ 





- +
¶
- ¶
+
¶
- ¶
+ 





¶
- ¶
+
¶
- ¶
=
n
1 i
i i i
n
1 i
n
1 j
i i i i
j
i
i j j i
i
i i
n
1 i
n
1 j
j
i
i j j i
i
i i
y
z
) fa cd (
) e a d b ( y
y
) e a d b (
y
) f b c e (
y
x
) d a d a (
x
) f a c d (
T   div   ) 37 . 6 (
 
(38) wj(Wj)=ajej-bjdj 
The condition of integrability is: 
(39) ∑ ∑ ∑
= = =
j e - e + x 





- w + e
n
1 j
j j j
n
1 j
j j j
n
1 j
j j j Y )  W div ( Y )  Z div ( T   div ) W ( ) 1 2 ( ÎSpan(X,Z) 
and with (35)-(38) becomes: 
) Z , X ( Span
z y
) fb (ce
x
) cd (fa
) d b a e )( 1 ( 2      
y z
) fb (ce
y
) b e e (b
x
) b d e (a
            
x z
) cd (fa
y
) d b e (a
x
) d a d (a
-   ) 40 . 6 (
n
1 j
j
j j
n
1 j
j
j j
n
1 j
j j j j j
n
1 j
j
j j
n
1 i
i
j i j i
n
1 i
i
j i j i
n
1 j
j
j j
n
1 i
i
j i i j
n
1 i
i
j i i j
Î
¶
¶






¶
- ¶
-
¶
- ¶
+ - + e
+
¶
¶






¶
- ¶
+
¶
- ¶
+
¶
- ¶
+
¶
¶






¶
- ¶
+
¶
- ¶
+
¶
- ¶
∑ ∑ ∑
∑ ∑ ∑
∑ ∑ ∑
= = =
= = =
= = =
 
  The conditions that g(X,X)=0, g(Z,Z)¹0, g(X,Z)=0 become: ACTA UNIVERSITATIS DANUBIUS                                                       Nr. 1/2005 
 
  38 
(41) 









= + e + 





- 




 -
¹ e + e + 




 -
= e + e + 




 -
∑ ∑ ∑
∑ ∑ ∑
∑ ∑ ∑
= = =
= = =
= = =
0 ) e b d a ( f y d c y a
0 e d f y d
0 b a c y a
n
1 i
i i i i i
n
1 j
j
j
n
1 i
i
i
n
1 i
2
i i
n
1 i
2
i i
2 n
1 i
i
i
n
1 i
2
i i
n
1 i
2
i i
2 n
1 i
i
i
 
  From (40) we have: 
2
y
) fb (ce
x
) cd (fa
) d b a e )( 1 ( 2 f c
z
) fb (ce
y
) b e e (b
x
) b d e (a
e b
z
) cd (fa
y
) d b e (a
x
) d a d (a
d a
  rank
n
1 p
p
p p
n
1 p
p
p p
n
1 p
p p p p p
k k
n
1 i
i
k i k i
n
1 i
i
k i k i
k k
j j
n
1 i
i
j i i j
n
1 i
i
j i i j
j j
=


























¶
- ¶
-
¶
- ¶
+ - + e






¶
- ¶
+
¶
- ¶
+
¶
- ¶






¶
- ¶
+
¶
- ¶
+
¶
- ¶
-
∑ ∑ ∑
∑ ∑
∑ ∑
= = =
= =
= =
 
where j, k=1,...,n. 
 
  We have now: 
0
y
) fb (ce
x
) cd (fa
) d b a e )( 1 ( 2 f c
z
) fb (ce
y
) b e e (b
x
) b d e (a
e b
z
) cd (fa
y
) d b e (a
x
) d a d (a
d a
n
1 p
p
p p
n
1 p
p
p p
n
1 p
p p p p p
k k
n
1 i
i
k i k i
n
1 i
i
k i k i
k k
j j
n
1 i
i
j i i j
n
1 i
i
j i i j
j j
=






¶
- ¶
-
¶
- ¶
+ - + e






¶
- ¶
+
¶
- ¶
+
¶
- ¶






¶
- ¶
+
¶
- ¶
+
¶
- ¶
-
∑ ∑ ∑
∑ ∑
∑ ∑
= = =
= =
= =
 
therefore: 
0 ) d b e a (
y
) fb (ce
x
) cd (fa
) d b a e )( 1 ( 2
) fa cd (
z
) fb (ce
y
) b e e (b
x
) b d e (a
) fb ce (
z
) cd (fa
y
) d b e (a
x
) d a d (a
j k k j
n
1 p
p
p p
n
1 p
p
p p
n
1 p
p p p p p
j j
k k
n
1 i
i
k i k i
n
1 i
i
k i k i
k k
j j
n
1 i
i
j i i j
n
1 i
i
j i i j
= - 





¶
- ¶
-
¶
- ¶
+ - + e
+ - 





¶
- ¶
+
¶
- ¶
+
¶
- ¶
+ - 





¶
- ¶
+
¶
- ¶
+
¶
- ¶
∑ ∑ ∑
∑ ∑
∑ ∑
= = =
= =
= =
 
 ŒCONOMICA 
 
  39
0
y
) fb (ce
) d b e a (
x
) cd (fa
) d b e a ( ) d b a e )( 1 ( 2 ) d b e a (
z
) fb (ce
) fa cd (
y
) b e e (b
) fa cd (
x
) b d e (a
) fa cd (
z
) cd (fa
) fb ce (
y
) d b e (a
) fb ce (
x
) d a d (a
) fb ce (
n
1 p
p
p p
j k k j
n
1 p
p
p p
j k k j
n
1 p
p p p p p j k k j
k k
j j
n
1 i
i
k i k i
j j
n
1 i
i
k i k i
j j
j j
k k
n
1 i
i
j i i j
k k
n
1 i
i
j i i j
k k
=
¶
- ¶
-
-
¶
- ¶
- + - + e -
+
¶
- ¶
- +
¶
- ¶
-
+
¶
- ¶
- +
¶
- ¶
-
+
¶
- ¶
- +
¶
- ¶
-
∑
∑ ∑
∑
∑
∑ ∑
=
= =
=
=
= =
 
 
0
y
) fb (ce
) d b e a (
x
) cd (fa
) d b e a ( ) d b a e )( 1 ( 2 ) d b e a (
y
) b e e (b
) fa cd (
x
) b d e (a
) cd fa (
z
fb ce
cd fa
) fb (ce
y
) d b e (a
) fb ce (
x
) d a d (a
) fb ce (
n
1 p
p
p p
j k k j
n
1 p
p
p p
j k k j
n
1 p
p p p p p j k k j
n
1 i
i
k i k i
j j
n
1 i
i
k i k i
j j
k k
j j
2
k k
n
1 i
i
j i i j
k k
n
1 i
i
j i i j
k k
=
¶
- ¶
-
-
¶
- ¶
- + - + e -
+
¶
- ¶
-
+
¶
- ¶
- -
¶
-
-
¶
- +
+
¶
- ¶
- +
¶
- ¶
-
∑
∑ ∑
∑
∑
∑ ∑
=
= =
=
=
= =
 
0
y
) fb (ce
) d b e a (
y
) d b e (a
) fb ce (
x
) b d e (a
) cd fa (
x
) cd (fa
) d b e a (
) d b a e )( 1 ( 2 ) d b e a (
y
) b e e (b
) fa cd (
z
fb ce
cd fa
) fb (ce
x
) d a d (a
) fb ce (
n
1 p i,
p
p p
i k k i ij i
j i i j
p p kp
i
k i k i
p p jp
n
1 p , i
p
p p
i k k i ij
n
1 p
p p p p p j k k j
n
1 i
i
k i k i
j j
k k
j j
2
k k
n
1 i
i
j i i j
k k
=
¶
- ¶
- d -
¶
- ¶
- d +
¶
- ¶
- d -
¶
- ¶
- d
+ - + e -
+
¶
- ¶
- +
¶
-
-
¶
- +
¶
- ¶
-
∑
∑
∑
∑ ∑
=
=
=
= =
 
From (38) we have: 
(39)    0 ) ce bf (
z
ce bf
af cd
) ce bf (
y
ce bf
ae bd
) ae bd (
x
ae bd
af cd
2 2
1
2
1 = -
¶
-
-
¶
+ -
¶
-
-
¶
+ -
¶
-
-
¶
 
   ACTA UNIVERSITATIS DANUBIUS                                                       Nr. 1/2005 
 
  40 
From (39), (40) we have: 
(41) 
a y ) c ay (
c ay
ce bf
ae bd
1 1
1
- -
-
=
-
-  
(42) 
c ay
1
ae bd
af cd
1 -
- =
-
-  
(43) 
a y ) c ay (
1
ce bf
af cd
1 1 - -
-
=
-
-  
  If we replace (41)-(43) in (39) we have: 
(44)  0 c a
z
a
y
a
c
y
c
a
x
c
y ac 2
z
c
x
a
) y ( a
z
a 2 2
1 1 1
1
1
2 1 2 = 





- -
¶
¶
-
¶
¶
-
¶
¶
+
¶
¶
- +  

 
 +
¶
¶
-
¶
¶
+  

 
 -
¶
¶  
  If we note A=ay
1-c we have: 
(45)  0
z
a
z
A
y A
y
A
a
y
a
A
x
A 1 2
1 1 1 = 





¶
¶
-
¶
¶
+  


 


-
¶
¶
-
¶
¶
+
¶
¶  
  We have. 
div X= 







+
+
+
¶
¶
+ 







+
+
+
¶
¶
+ 







+
¶
¶ ∑ ∑ ∑ ∑ ∑
+
=
+
=
+
=
+
=
+
=
1 n 2
1 j
j
1 n 2 n
1 a
1 n 2
1 j
j
a
a n n
1 i
1 n 2
1 j
j
i
i
X
j   1 n 2
1 n 2
z
X
X
j   a n
a n
y
X
X
j   i
i
x
X  
 
REFERENCES: 
1  Bejancu, A., Duggal, K. L., Lightlike Submanifolds of Semi-Riemannian Manifolds and 
Applications, Dordrecht, Kluwer Acad.Publishers, 1996; 
2  Blair,  D.  E.,  Contact  Manifolds  in  Riemannian  Geometry,  Springer-Verlag,  Berlin-
Heidelberg, New York, 1976; 
3  Chen, B. Y., Geometry of Submanifolds, New York, Marcel Dekker, 1973; 
4  Eisenhart, L. P., Riemannian Geometry, Princeton University Press, 1926; 
5  Ioan,  C.  A.,  Totally  Umbilical  Lightlike  Submanifolds,  Bull.  Math.  Soc.  Sc.  Math. 
Roumanie, Tome 37 (87), No. 1-4, 1996; 
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7  Ioan,  C.  A.,  Totally  Geodesic  Foliations  and  Minimal  Foliations  on  the  Semi-
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Vol.58, No.1 (1997); 
9  Kobayashi,  S.,  Nomizu,  K.,  Foundations  of  Differential  Geometry,  New  York, 
Interscience Publishers, vol. I, 1963, vol. I, 1969; 
10  Lawson, H. B., Foliations, Bull. Amer. Math. Soc., (80), 3, 1974; 
11  Molino, P., Riemannian foliations, Boston, Basel, Birkhauser, 1988; 
12  O'Neill,  B.,  Semi-Riemannian  Geometry  with  Applications  to  Relativity,  Academic 
Press, 1983; 
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Contact Manifolds in Riemannian Geometry,

Degenerate Submanifolds of Semi-Riemannian Manifolds,

Differential Geometry of Foliations,

Foundations of Differential Geometry,

Geometry of Submanifolds,

Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications,

Riemannian foliations,

Riemannian Geometry,

Sasakian Manifold with Pseudo-Riemannian Metric,

Semi-Riemannian Geometry with Applications to Relativity,

Totally Geodesic Foliations and Minimal Foliations on the SemiRiemannian Manifolds,

Totally Geodesic Foliations on Semi-Riemannian Manifolds,

Totally Umbilical Lightlike Submanifolds,

Degenerate Foliations in Sasakian Semi-Riemannian Manifolds

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Degenerate Foliations in Sasakian Semi-Riemannian Manifolds

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