Tikrit Journal of Pure Science 20 (4) 2015
ISSN: 1813 - 1662
Global Conharmonic Concept Type Nearly Kahler Manifold
Ali A. Shihab
Department of Mathematics , college of Education for pure sciences , University of Tikrit , Tikrit , Iraq
[email protected]
Abstract
The concept of permanence conharmonic type Nearly Kahler manifold conditions are obtained when the Nearly
Kahler manifold is a manifold conharmonic constant type. Proved that the local conharmonic constancy of type
Nearly Kahler manifold is equivalent to its global conharmonic constancy of type. And also proved that the
Nearly Kahler manifold conharmonic constant type is a manifold of constant scalar curvature.
The notion of constancy of type Nearly Kahler manifolds was introduced A.Greem ([1], [2], [3]) and has proved
very useful in the study of the geometry of nearly kahler manifolds. Next nearly kahler manifolds of constant
type considered various mathematicians (see. [1], [3], [4], [5], [6], [7]). Comprehensive description of Nearly
Kahler manifolds of constant type was obtained V.F.Kirichenko [6]. In [6] it is shown that Nearly Kahler
manifold of pointwise constant type description by the identity
, where B - a function on
cd
Babh B hcd  B ab
Nearly Kahler manifold, cd
. Wherein local constancy of type Nearly Kahler manifold is
 ab   ac bd   ad  bc
equivalent to the local constancy of its type. Moreover, the covariant constancy of structure tensors of the first
and second kind it follows immediately that, the constancy of type at only one point Nearly Kahler space implies
global consistency of its type.
Keywords: Conharmonic concept type nearly kahler manifold, manifold concept scalar curvature.
Recall the definitions
Let 𝑀2𝑛 - Nearly Kahler manifold with АН structure {g, J}  - Riemannian connection of the
metric g.
Definition 1. [3]. Nearly Kahler manifold M2n has
constant type at a point рM2n if X,Y,ZT(M2n):
<X,Y>=<JX,Y>=<X,Z>=0;
(Here T(M2n ) Y  Z   J Y    J Z  .
X
Theorem 1. Nearly Kahler manifold is a manifold
conharmonic constant type if and only if
B abh Bhcd 




1
{ da Acb  3Bcb   ca Adb  3Bdb 
4(n  1)




1 ab
  cb Ada  3Bda   db Aca  3Bca }   c cd
.
2
The proof:
Let 𝑀2𝑛 - Nerely Kahler manifold. To calculate the
(X, Y, Z, W) and K (X, Y, JX, JY) on the space of the
associated G-structure:
ˆ
K  X , Y , Z ,W   K ijkl X iY j Z kW l  K aˆbcdˆ X aˆ Y b Z cW d 
X
tangent space 𝑀2𝑛 in рM2n.)
Definition 2 [3]. Nearly Kahler -manifold having a
constant type at each point, called NK-manifold of
pointwise constant type.
Definition 3 [3]. Nerely Kahler manifold 𝑀2𝑛
pointwise constant type is globally constant type if
X,YХ(M2n):
<X,Y>=<JX,Y>=0;
.
X  Y  1   X J Y   const
These concepts have been summarized L. Vanhecke
and Boutenom for arbitrary almost Hermitian
manifolds [8].
As shown by L. Vanhecke, if the almost Hermitian
manifold belongs to Nearly Kahler manifolds, these
definitions are reduced to the above definitions
A.Greya.
It is now natural to introduce the definition of
constant type, similar definitions Vanhecke, replacing
the tensor R Riemann-Christoffel tensor conharmonic
curvature.
We recall some useful formulas, we need, is easily
obtained from the first group of structural equations
Nearly Kahler manifolds [6], [7]:
ah
ah c
ahc
1) dAba  dAbh
 Abhc
  Abh
c  Aca bc  Abс ca ;
(2)
ad
hd a
ah d
ad h
ad h
2) dBbc
  Bbc
 h  Bbc
 h  Bhc
 b  Bbh
c ;
ˆ
ˆ
ˆ
 K aˆbcˆd X aˆ Y b Z cˆW d  K aˆbˆcd X aˆ Y b Z cW d  K abˆcdˆ X aY b Z cW d 
ˆ
ˆ
ˆ
 K abˆcdˆ X aY b Z cW d  K abcˆdˆ X aY b Z cˆW d ;
ˆ
K  X , Y , JX , JY   K ijkl X iY j  JX k  JY l  K aˆbcdˆ X aˆ Y b X cY d 
ˆ
ˆ
ˆ
 K aˆbcˆd X aˆ Y b X cˆY d  K aˆbˆcd X aˆ Y b X cY d  K abˆcdˆ X aY b X cY d 
ˆ
ˆ
 K abˆcˆd X aY b X cˆY d  K abcˆdˆ X aY b X cˆY d .
We show,
ˆ
K  X , Y , JX , JY   K  X , Y , X , Y   4 K aˆbˆcd X aˆ Y b X cY d
In this way,. Given the specter tensor of curvature
conharmonic last equation can be written as [9]:
ab
K  X , Y , JX , JY   K  X , Y , X , Y   [8Bcd


 
 


2
{ a Acb  3Bcb 
n 1 d

ˆ
  ca Adb  3Bdb   cb Ada  3Bda   db Aca  3Bca }] X aˆ Y b X cY d .
Since
ab
 8Bcd

cX

2
Y
2
ab c d
 4c cd
X Y X aYb
 
 
ab
  db Aca  3Bca }  4c cd
,
177

2
{ a Acb  3Bcb   ca Adb  3Bdb   cb Ada  3Bda 
n 1 d
ie
3) dBba  Bca bc  Bbc ca .
then
Tikrit Journal of Pure Science 20 (4) 2015
B abh Bhcd 



ISSN: 1813 - 1662
 

Since, c  C  M  dc (where, d  с    dc ) is a
horizontal form, and dc  ca a  c a a
therefore.
Then (3) can be written as
1
{ da Acb  3Bcb   ca Adb  3Bdb 
4(n  1)




1 ab
  cb Ada  3Bda   db Aca  3Bca }   c cd
.
2
 B fdh Babh cf  B cfh Babh df  B cdh B fbh af  B cdh Bbfh bf 
(1)
This theorem proved
Theorem 2. Local conharmonic constant type Nearly
Kahler manifold is equivalent to its global
conharmonic constant type.
The proof:
We differentiate externally with regard to (2)
Equation (1):
(3)
 B fdh B  c  B cfh B  d  B cdh B  f  B cdh B  f 
abh f
1
 ac
4n  1
1

d
4n  1 b
1

 ad
4n  1
1

c
4n  1 b

abh f
fbh a

A

A

A

A
1
dhg
dh
 ac Abhg
 g  Abh
g 
4n  1
1
chg
ch

 bd Aahg
 g  Aah
g 
4n  1
1
chg
ch

 ad Abhg
 g  Abh
g 
4n  1
1
dhg
dh

 bc Aahg
 g  Aah
g 
4n  1
1 cd
   ab
сg g  c g g .
2

bfh b

dhg
dh g
Abhg
  Abh
 g  Adf  3B df  bf  Abf  3Bbf  df 
chg
ch
Aahg
 g  Aah
 g  Acf  3B cf  af  Aaf  3Baf  cf 
chg
ch g
Abhg
  Abh
 g  Acf  3B cf  bf  Abf  3Bbf  cf 
dhg
dh g
Aahg
  Aah
 g  Adf  3B df  af  Aaf  3Baf  df 
d
f


 


 


 


 
 3B df  bf  Abf  3Bbf  df 
c
f
 3B cf  af  Aaf  3Baf  cf 
c
f
 3B cf  bf  Abf  3Bbf  cf 
d
f
 3B df  af  Aaf  3Baf  df 

The last equality can be rewritten as:
1 cd
   ab
dc.
2
 


 


 


 

 
 
 
 

 
 
 
 




1
  B fdh Babh 
 bd Aaf  3Baf   af Abd  3Bbd   ad Abf  3Bbf   bf Aad  3Bad  cf 
4
(
n

1
)


 cfh

1
  B Babh 
 ac Abf  3Bbf   bf Aac  3Bac   bc Aaf  3Baf   af Abc  3Bbc  df 
4(n  1)





1
  B cdh B fbh 
 bd Acf  3B cf   cf Abd  3Bbd   df Abc  3Bbc   bc Adf  3B df
4
(
n
 1)


1
  B cdh Bafh 
 ac Adf  3B df   df Aac  3Bac   cf Aad  3Bad   ad Acf  3B cf
4
(
n
 1)

 


  

 
 
  
 
 

f
a 

 f
 b 

1
 af Abd  3Bbd   bf Aad  3Bad  cf   bf Aac  3Bac   af Abc  3Bbc  df 
4(n  1)
1

 cf Abd  3Bbd   df Abc  3Bbc  af   df Aac  3Bac   cf Aad  3Bad  bf 
4(n  1)
1
dg
cg
cg
dg
dgh
cgh
cgh
dgh

 ac Abgh
  bd Aagh
  ad Abgh
  bc Aagh
 h   ac Abg
  bd Aag
  ad Abg
  bc Aag
h 
4n  1
1 cd
   ab
c h h  c h h ,
2


 
 

 

which in view of (1) takes the form

 
 
c fd c c cf d c cd f c cd f
1
 ab  f   ab f   fb  a   af  b 
Abd  3Bbd  ac  Aad  3Bad  bc 
2
2
2
2
4(n  1)
1

Aac  3Bac  bd  Abc  3Bbc  ad 
4(n  1)
1

Abd  3Bbd  ac  Abc  3Bbc  ad  Aac  3Bac  bd  Aad  3Bad  bc 
4(n  1)
1
dg
cg
cg
dg
dgh
cgh
cgh
dgh

 ac Abgh
  bd Aagh
  ad Abgh
  bc Aagh
 h   ac Abg
  bd Aag
  ad Abg
  bc Aag
h 

4 n  1
1 cd
   ab
c h h  c h h ,
2

 
 

 
 





 

 

and we have
1
cg
dg
dg
cg
1)
 ad Abgh
  bc Aagh
  ac Abgh
  bd Aagh
  abcd c h ;
2n  1
1
2)
 ad Abgcgh   bc Aagdgh   ac Abgdgh   bd Aagcgh   abcd c h .
2n  1
… (4)


From (4) we have



1
cg
dg
dg
cg
 ad Abgh
  bc Aagh
  ac Abgh
  bd Aagh
  abcd c h ;
2n  1
1
2)
 ad Abgcgh   bc Aagdgh   ac Abgdgh   bd Aagcgh   abcd c h .
2n  1
1)



Rolling recent last equality in the first by indices a
and c, and then the indices b and d, we obtain:
178
Tikrit Journal of Pure Science 20 (4) 2015
1) ch 
1
ab
Aabh
;
nn  1
2) c h 
ISSN: 1813 - 1662
1
Aababh. (5)
nn  1
The proof:
In particular, from (6) it follows that. From
ab
abh
Aabh
 Aab
 0 this and from (2: 1) it follows that
Rolling equation (4), first in the indices a and d, and
then the indices b and c, we get:
(6)
1
1
ab
h
abh
1) ch  
nn  1
Aabh ;
2) c  
nn  1
ab
ab c
abc
dA  dAaa  dAab
 Aabc
  Aab
c  Aca ac  Aac ca  0 .
5
From (2: 3), we have dB  dBaa  Bca ac  Bac ca  0 . So
from the last two equations, we have that
d   2dA  6dB  0 , that   const .
This theorem proved.
Aab .
Comparing (5) and (6) we find that: ch  c h  0 .
This theorem proved.
Theorem 3. Nearly Kahler manifold conharmonic
constant type is a manifold of constant scalar
curvature.
References
1. Gray A. Nearly Kâhler manifolds. J. Diff. Geom.,
4, №3, 1970, 283-309.
2. Gray A. The structures of nearly Kähler manifolds.
Ann. Math. 223 (1976), 233-248.
3. Gray A. Six dimensional almost complex
manifolds defined by means of three-fold vector cross
products. Tôhoku Math. J., 2, 1969, 614-620.
4. VF Kirichenko Some types of K-spaces .// Uspekhi
Mat. Sciences, T.30, №3, 1975, p. 163-164.
5. VF Kirichenko Some results of the theory of kspace. 6th All-Union. Geometry. Conference on the
present-day. probl. geometry. Abstracts. Vilnius,
1975, 112-115.
6. VF Kirichenko K-spaces of constant type. Sib.
Mat. g., 1976, t.17, №3, 282-289.
7. VF Kirichenko K-algebra and K-spaces of constant
type with indefinite metric .// Mateo. notes, 29, №2,
1981, 265-278.
8. Vanhecke L., Bouten F. Constant type for almost
Hermitian manifolds .// Bull. Math. Soc. Sci. math.
RSR. - 1976 (1977), v.20, №3-4, p. 252-258.
9. VF Kirichenko, Rustanov AR, Shihab A.
Geometry curvature tensor conharmonic almost
Hermitian manifolds .// Mat. Notes, 2011, t. 90, №1.
‫نوع الثابت العالمي الكونهورمني لمنطوي كوهلر التقريبي‬
‫علي عبد المجيد شهاب‬
‫ العراق‬، ‫ تكريت‬، ‫ جامعة تكريت‬، ‫ كلية التربية للعلوم الصرفة‬، ‫قسم الرياضيات‬
‫الملخص‬
‫ حيث تم برهان‬.‫الثابت الكونهورمني لمنطوي كوهلر التقريبي والذي يتحقق عندما منطوي كوهلر التقريبي يكون منطوي كونهورمني ذات ثابت معين‬
‫ وبرهن ايضا كل منطوي كوهلر‬.‫الثابت المحلي الكونهورمني من نوع المنطوي كوهلر التقريبي يكون مساوي الى نوع الثابت المحلي الكونهورمني‬
‫التقريبي ذات ثابت نوع كونهورمني هو منطوي ذات ثابت منحني عددي‬
179