On complex conformal connections in an almost complex manifold with a torsion tensor

J.

Vol. 16, No. 1, 1979

ON COMPLEX CONFORMAL CONNECTIONS IN AN ALMOST COMPLEX MANIFOLD WITH A TORSION TENSOR

By OK KYUNG YOON

§ 1. Introduetion

Let M be an n-dimensional Riemannian manifold with metric tensor gji·

The change of the metric

gji=e'#gji,

where p is a certain scalar function, does not change the angle between two vectors at a point and so is called a conformal change of the metric.

If there exists a function p such that the Riemannian manifold with me- tric tensor e2

ji is locally Euclidean, the Rimannian manifold is said to be .conformally flat.

It is well known (Weyl DJ) that the so-called Weyl conformal curvature tensor

Wjjik . Kjjik+oiCji-O/'Cji+Cjkgji -C/gki

is invariant under a conformal change of g, where Kkjik is the Riemann- Christoffel curvature tensor of M and

( 1) (

11

Cji

n-2 K ji +· 2(#-1) (n-2) Kgji, Ci=CksgBk, K

KsJi', K=gjiKji

and a necessary and sufficient condition for M to be conformally flat is that Wkjt=O for n>3

and

P'kCji-P'jCji=O for .n=3,

P

denoting the 'operator '0£ covariant:differentiation with respect to Chris- toffel symbols formed with

A complex analogue of the above in a .Kaehler manifold is given by K.

Yano (K. Yano [2J).

ReceivedMay 7. 1979

72 Ok KyungYoon

In a Kaehler manifold M with the Hermitian metric tensor gji and com- plex structure tensor Fih, we have

17

I7 kF,h=O, J7 kFji =O,

"Wh~n

Fji=F/gsi and consequently Fji = -Fijo The affine connection which satisfies

Df/!2P

gji =O,

Fji =O and torsion tensor

1/2 _{(Fjih_Fi})}

-Ejiqh,

-where p is a scalar function and qh is a vector field, is given by F ..h

{h}

^-l-p.

.h+p .o.h -

g ..+q .F.h+q.F.h -qhF ..

-where

Pi=OiP, ph=PsgFh, qi= -PsF/, if=qsgsh.

For this connection called a complex conformal connection, K. Yano proved the following theorem:

If in an n-dimensional Kaehler manifold (n24) , there exists a scalar func- .lion P such that the complex conformal connection is of zero curvature, then the

-Bochner curvature tensor of the manifold vanishes.

Define C"jih by

Ckjih=K"jih+oiLji-O!Lki+gjiL"h_g"iLl + F"hM··-F.kM".+MkkF··-M·hF,,·

+( n~4 ) VkjFik+F"jB,h, where

V"j = 1/ 2K"j/Ft', Mji = -KjsF/,

Lji=MjsF/,

L"h=L"sgsh, M"k=M"sgsh, Bih=Bisg'h,

Bji=Hji+nMji-2Mij- n~2 (n~4 V+l/2K )Fji,

V = VstFst and K = Kstgst.

This tensor is called a complex conformal curvature tensor. The complex conformal curvature tensor Ckjih is invariant under a change of the complex conformal connection. (0. K. Yoon [5J).

A complex analogue of the above in an almost complex manifold with a torsion tensor is not yet known. The main purpose of the present paper is to try to find some properties concerning this problem. In § 2 we state some of fundamental formulas in an almost complex manifold with a torsion tensor to fix our notation and in § 3, we introduce what we call complex conformal connections in an almost complex manifold with a torsion tensor.

In § 4, we study the integrability condition for an almost complex ma- nifold M with a torsion tensor, in § 5, we state some of fundamental for- mulas, and in § 6 we study an invariant curvature tensor.

§ 2. Preliminaries

We consider an n-dimensional almost complex manifold M with a torsion tensor Sjih=I!2(rjih-rii) which is covered by a system of coordinate neighborhoods

and denote by gji and Fih the components of the Hermitian metric tensor and those of the complex structure tensor of M re- spectively, where here and in the sequel the indices

run over the range {I, 2, "', n}.

We denote by Dj the operator of covariant differentiation with respect to rjih, then the torsion tensor Sjih=1!2(rjih-ri/) is given by

(2.1) S··h=-uhF··

J' J"

where u

are components of a vector field. Then, we have

(2.2) Dkgji=O, DkFl=O, DkFji=O,

where Fji=F/gsi is skew symmetric.

Above all, we notice that an affine connection is symmetric, that is.

which satisfies (2.3)

and whose torsion tensor

(2.4) 1/2(r

h- r.}) =Sjih

is uniquely determined and given by

(2.5) r··h=

{h}

+S..h+S..h+S..h

J' J' 'J ,

where (Hayden [6J)

(2.6)

74

So we have, for the components r

of affine connection in an almost complex manifold M with a torsion tensor

(2.7) r

{ji} +UjF~+UiF!-uhFji'

where {;} are the Christoffel symbols formed with

and

Fhj=ghsFsj = -Fjs!th =

-Ft,

We denote by (2.8)

the components of the Riemann-Ghristoffel curvature tensor of M, where

Qk=a/a~k.

It is well known that

and

satisfy (2.9)

(2.10) (2.11) and

(2.12) (2.13) (2.14) where

K kjih= - K kjih, Kkjih= - Kkjhi, Kkjih= Khijk,

K kjih + Kjikh+ Kikjh=O

17eKkjih+17kKjeih+17jKekih=O,

17sKkj/=17kKji-17jKki,

2I1

K

K,

Kji=Kij=KSj/

and

are the Ricci tensor and the scalar curvature symbol {jh _i ^{} ,} respectively.

§ 3. Integrability conditions

We consider the integrability conditions in an almost complex manifold M with a torsion tensor.

From (2.7), using

we have

DkFji=fi'kFji-uigkj+Ujgki-WjFki+WiFkj=O,

where

consequently

(3.1)

Then, from the Ricci identity, we obtain

(3.2) 'l7 _z 'l7 _{i Fj} ;-f iFzFj;=KzijsF/-KliisFl

= gijUU - gljUii"- gkiUlj+ guUkj - FijWU+ F1jWii+ Fk;Wlj - FUWij>

where (3.3) (3.4) (3.5)

If we define (3.6)

(3.7)

Uii=V'jUi-UjWi+ 1/2pFji, Wj;=V'jWi-WjWi+ 1!2pgji= -UksF/,

p=usUtgst=wsWtgst•

KsjitFst=Aji, KjsFl=-Hji then, from (2.9), (2.10) and (2.11), we have

(3.8)

And, from (3. 2) , we have also (3.9)

consequently (3.10) (3.11) (3.12)

2Aji - (Hj;- H ij) = (n-4) (Uji -Uij) + 2 (ustF't) F ji, H _{ji +} H

= - (n-2) (Uji+Uij),

A-K=2(n-2) (UstFst ) ,

(3.13)

where A

AstFst, K=Kstgst

H st Fst.

From (3. 10) , (3. 11) and (3.12), we find Uji=- 2(n~2) ^(Hji + ^{H ij)}

(3.14)

On the other hand, since the Nijenhuis tensor

N ..

h=F.s{'=' F·h_'='.F h) -F.s(;} F.h_;}.F h)

s .

s • s

s

76 Ok KyungYoon

of the complex structure tensor Fik vanishes in virtue of (3. 1), the complex structure tensor Fik be integrable.

Thus, we have

PROPOSITION

3. 1. In an n-dimensional Riemannian manifold

(n>4, even number) with metric tensor gji, the necessary and sufficient condition such that the manifold M can be admissible an almost complex structure Fik which satisfies

Dkgji=O, DkFji=O and Fjik_Fil= -2Fjiu

is follows;

Kkjik

Kjkih, Kkjih

Kkjhi, Kkjih

K hijk, Kkjik+Kjikk+ Kikjk=O,

fllKkjt+flkKjUk+fljKlkt=O and K1kjsF/ - KlkisK/ = gkjUU - gljUki - gkiUlj+ gUUkj

- FkjWu+ FljWki+ FkiWlj - FliWkj, where, Uji

If n=4, the last condition can be replaced by

where

and

§ 4. Some formulas in an almost complex manifold M with a t.or- sion tensor

We denote by

(4.1) Rk··k=O.F ·.k-o.r•.k+F•.kF ..s-F· kFk·s

the components of the curvature tensor of M.

By a straightforward computation, from (2. 7), we find

(4.3) (4.4)

(4.2) Rkjjh=Kkjl-FkhUji+F/Uki-U'/IFjj+u/F/d

+Fl(Vkj-Ujk-pFkj) -2Fkj(Uiwh-uhwi) + Uk (O/Uj - gjjU

+ F/Wi - Fjiwh+ Flwj) -Uj(O,/IUj - gkjUh+ FkhWj - FkiWh+ Fihwk) , where wh=wsg'h, from which Rkjjh=Rkj/gsk satisfy

Rkjill

Rjkih, Rkjik = - Rkjki, Rkjjk - Rkijk

=-~X~+~Xoo-~X~+~~~

+ F jhX [kj] - FkjX [ik] ,

(4.5) R/<jjk+ Rjikk+ Rikjk= -2FkjXjh-2FjjXkk-2FikXjk, where X ejiJ =Xji+Xij, X

=Xjj-Xjj,

and

(4.6) X ji =Uji+2ujwj-ujWj+ (0/2) Fji•

Using DkFjj=O, from the Ricci identity, we have (4.7)

Now, if we define

(4.8) RstjiFst= -2Ejj , RjistFst= -2 Vji, RsjitFst=Aji, then, from (4.4) and (4.5), we have

(4.9) (4.10) (4.11)

where X=XstFst,

then we have

2 Vji -2Ejj = -nXjj +2XFji, A jj -Aij -2 Vjj=2Xji-2XFji'

Aji-Ejj=- (n-2)Xji, and consequently if we put

A = AstFst, E= EstFst (

VstFst) ,

(4.12) x _{n-2 (E-A).} ¹

From (4. 11) , we have (4.13) X [jj] 1

2 {2Ejj-(Ajj-Ajj)}, X ejiJ =- 1

2 (Aji+A ij).

n- n-

78 Ok Kyung Yoon

-furthermore, eliminating Xji from (4.9), (4.10) and (4.11) we have

·(4.14) n(Aji-Ai) -4Eji -2(n-Z) Vji+Z(E-A) Fji=O.

Substituting (4.11) and (4.13) into (4.4) and (4.4), we have 0(4. 15)

{4.16) Rkjih+ Rjikh+ Rikjh

n:'Z {Fkj(Aih-Ein) +Fji(Akh-Ekn) +Fik(Ajh-Ejh)}.

On the other hand, from the Bianchi identity, we have (4.17) DtRkjih+ DkRjlih + DjRwh

=2u

(FlkRsjih + FkjRslih + FjlRskih) ' hence, by contracting with Fkj, we obtain

(4.18)

Substituting (4.18) into (4.17), we have (4.19) DtRkjih + ^DkRjlih + ^{D jRlkih}

n~2 {Flk(PStDsRtju,-DjEin) +Fkj(FstDsRtlih-DtEih) + F jl (PStDsRtkih - DkEih) }•

Therefore, we have the following

PROPOSITION 4.1. In an n-dimensional

.almost complex manifold M with Hermitian metric tensor gji, complex structure tensor Ft, the affine con- nection which satisfies

and

where u

a vector field, is given by

Pjih={;;} +F/'ui+Fi!'uj-Fjiuh.-

Furthermore, for the curvature tensor, the following relations hold;

Rkjih=-Rjkih, Rkjih=-Rkjhi, RkjisFhS = RkjhsF/,

Rkjih+ R jikh + ^Rikjh

n':'2 {Fkj(Aih-Eih) + ^{Fji (A}

^{E kh )} + Fik (A jh - E jh )} , nAji -4Ej;-2(n-2) V j ;+2(E-A)Fji =O

..and

D1Rkjih + ^DkRjlib. + ^{D jRlkih}

---=2- (F1kDsRtjih+ FkjDsRtlih+ FjlDsRtkih) pt 2 n-

§ 5. Complex conformal connections

In an almost complex manifold M with a torsion tensor S··h

= - u

F .,

we

<consider a conformal change of Hermitian metric

(5.1) Kji=e2Pgji, Fji=e2PFji, Fih=Fjsgsh=Fih,

where p is a scalar function, and we look for an affine connection f ji

such .that

.(5.2)

D"gji=O, DkFji=O,

where D" are the operator of covariant differentiation with respect to the con- nection Fjih, and the torsion tensor Sjih is given by

(5.3) Sjih= -vhFji,

where v

are components of a vector field. We call such a metric change a complex conformal change of the metric.

By the remark above, we have, for the components f jih of this affine connection,

(5.4)

where {;i} are the Christoffel symbols with gji, and Vi=VSgsi.

80

Also,

(5.5) we have (5.6)

Ok Kyung Yoon

where Pi =OiP and pk=Psgsh. From (2. 7), using (5. 1), we have (5.7) Fjih=Fjih+pjOih+P•.o/-phgji

+ (Vj-Uj) Fih+ (Vi-Ui)F/- (e2Pv h

h) Fji- If we define qh by

(5.8) then

(5.9)

Substituting (5.8) and (5.9) into (5.7), we have

(5.10) F··h

= r··

h+p·o·h+p·o.h-

g ..+ q. F .h+q·F·h - ,.h.F.·

J'- We now compute 15

F ji and find

DkFji=Dk(e2PFji)

=e2P {(qsF/-pj) F ki + (Pi-qsF/)Fkj - (qi+PiF/)gkj+ (qj+PsF/)gkj}.

Thus, in order that DkFji=O, we must have

(qsF/-pj) F

+ (Pi-qsF/) Fkj- (qi+PsF;,)gkj+ (qj+PsF/)gki=() for which, transvecting with gkj, we find

(n-2) (PsF/+qi) =0, that is, assuming n::2: 4

Therefore, (5.11)

The converse being evident, we have

PROPOSITION

5. 1. In an almost complex manifold with a torsion tensor,.

JJy a complex conjormal change oj the metric gji=e2

gji, Fji =e

Fji, the affine connection Fjih which satisfies

15ijji=o, 15I<Fji =O

Fjih_Fil= -2vhFji,

where P is a scalar function and vh is a vector field, is given by F ..h

r·

.h+pJ.h +p·o·h- ....kg.. +q. F-h+ q .F-h _qh F ..

-Where

P .-O.p " - , , P - - s , .I.1t_P gsh q'= -p F.S ..

qh_ q gsh

We call such an affine connection a complex conformal connection m an :almost complex manifold with a torsion tensor.

§ 6. Curvature tensor of a complex conformal connection and its invariant

We consider a complex conformal connection (5. 10) in an almost complex :manifold with a torsion tensor and compute the curvature tensor of Fjih:

-(6.1) Rk··h=OkF··hh-o·Fk·h+Fk hF··s-F· hFk·s

By a straightforward computation, we find

(6.2) Rkj'=Rkjih+o/Pki-olpji- gjiPkh+ gkiPl

+ Flq/ei- Flqji- Fjiqkh+ Fkiql+ Fihakj-2F/ejf3ih, where

(6.3)

«6.4)

-(6.5) {6.6) {6.7) (6.8)

Pjj=DjPi-PjPi+qjqi+ (A/2)gji' Pji=Pij, A.=P#=qg,

al<j=qkj-qjk- flFkj, a/ej= -ajk.

fl=A+2Q.U',

82

(6.9)

If we define (6.10)

(6.11)

Ok Kyung"Yoon

f3ih={3isgsh, {3ji= -(3ij'

Rkj/gsh=Rkjih

RstjiFst= -2Eji, RjistFst= -2Vji

RSjitFst=Aji, then, from (4.8), we have

(6.12) Eji=Eji -2(qji-qij) - (a/2) Fji+n{3ji,

where a

astFst= 2p-nfJ., P=Pstgst=qstFst, and transvecting with

(=e2PFji), we have

e2PE=E-4P+ ~ (2f3-a), where a=astpt=2p-nfJ., {3={3stPt=2fJ. and

(6.13) e2PE=E- (n+4)p+ n(n+4) fJ..

2 From (4.8), (6.11) and (6.2), we have

(6.14) V ji = V ji -2(qji-qij) +2/.lFji - (n/2)aji' (6.15) Aji=Aji - (n-1)qji+qij-pFji-aji+2{3ji and transvecting with Fji (=e2PFii) , we have

(6.16) e2PV= V- (n+4)p+ n(n:4) fJ.,

(6.17) e2PA=A-2(n+1)p+ (n+4)/.l.

From (6.16) and (6.17), we find

(6.18) p= 2(n2~4) ^{(nA-2E) 2(n:-4)} (nA-2E)e2P~

(6.19) fJ.= (n+4)~n2-4) {(n+4)A-2(n+1)E}

(n+4) ~n2-4) {(n+4)A-2(n+1)E}e 2P . If we put

Q= 2(n:-4) (nA-2E) ,

N= (n+4)A-2(n+1)E, N=

(n+4) (n

-4)

2(n2~4) (nA-2E)~

(n+4)A-2(n+1)E

(n+4) (n2-4)

(6. 18) and (6. 19) can be written as

(6.20) P=Q-Qe2

p.=N-Ne2

and consequently

(6.21) PFji=QFji-QFji, P.Fji=NFji-N'Pji.

From (6.15), we have

(6.22) Aji=Aij=Aji-Aij-2(n+2) (qji-qij) - (2P-2p.) Fji+4{3ji, from which, using (6. 12) and (6.21), we have

(6.23)

(6.24)

1 -

qji-qij= (n+4) (n-2) {n(Aji-Aij) -4Eji -2(n-2)QFji}

(n+4)\n-2) {n(Aji-Aji) -4Eji -2(n-2)QFji},

{3ji 2(n+4~ (n-2) [4(Aji -Aij ) -2(n+2)ji

+ (n-2) {2Q- (n+4)N} Fji - ] 2(n+4~ (n-2) [4(Aji -Ai j ) -2(n+2)Eji+ (n-2) {2Q- (n+4)N} 'Pji]

On the other hand, from (6.15), we have

(6.25) Aji+Aij=Aji+Aij- (n-2) (qji+qij), from which, using (6.23), we find

(6.26) qji 2(n+4~ (n-2) {(n+4) (Aji+Aij)+n(Aji-Ai) -4Eji -2(n-2)QFji}

2(n+25 (n-2) {(n+4) (Aji + A ij ) +n(Aji - A ij ) -4Eji - 2(n-2)Q'Pj;}.

Substituting (6. 23) into (6.6), we obtain

(6 27) _. a _ji (n+4) (n-2) 1 [n(A ..

-A-.)

-4E··- (n-2) {2Q+ (n+4)N}

F ..

J

- (n+4~ (n-2) [n(A.ii-Aij) -4Eii - (n-2) {2Q+ (n+4)N} P ii]

84

{6.28)

Ok Kyung Yoon

M ji = 2(n+4) (n-2)

{(n+4)(Aji+Aij)+n(Aji-Aij) -4Eji-2 (n-2) QFji},

(6.29) Lji=MjsF;',

(6.30) Bji 2(n+4; (n-2) [4 (Aji-Aij) -2(n+2)Eji + (n-2) {2Q- (n+4)N} Fji].

(6.30) 7};= (n+4/(n-2) [n(Aji-A ij) -4Eji

- (n-2) {2Q+ (n+4)N} Fj;]

-then, (6. 24) , (6. 26) and (6. 27) can be written as

(6.32) f3ji=Bji -Bji, qji=Mji-Mji, aji= T.;i-Tji .and consequently

(6.33)

(6.34) L·"=L· Fs"

M·"=M· Fs"

B·"=B· Fs"

Substituting (6.32), (6.33) and (6.34) into (6.2), we have (6.35) Rkji"+0l'Lki -Ok"Lji - gj;Llt+ gkiLl

+ Fj"M/ti-Fk"Mji-FjiMk"+Fk;Ml +F/tTkj-2FkjB/t

=Rkji"+Oj"L/ti-Ok"Lji - gjiLi+ g/tiLl + F·" M.·- F/tM··- F··M."+ F.·M·"

+ Fi" Tkj - 2FkjB/t If we define Ckji" as

<6. 36) Ckji"

Rkj/t + O/'L ki - Ok"Lji - gjiLk"+ g/tiLj"

+ F/'Mk;- Fk"Mji - FjiMk"+ FkiM/' +F·"T,..-2F•.Bft

-then (6. 35) reduces into {6.37)

We call such a tensor Ckji" a complex conformal curvature tensor in an

almost complex manifold with a torsion tensor.

Thus, we have the following theorem.

THEOREM In an n-dimensional (n::::::4) almost complex manifold with a tor- sion tensor, the complex conformal curvature tensor C kji" defined by

Ckji"=Rkji"+olLki-Ok'ILji- gjiLk"+ gkiLl +F·"Mk·-Fk"M.·-Y·Mk"+Fk·M·"

+ Fi"Tkj-2Fk

Bi"

is an invariant under a complex conformal change of metric gji=e2

gji, F'ji=e

Fji.

Reference 1. ^H.

2- K.

3-

4.

5. O.

6. H.

Weyl, Zur Infinitesimal Geometrie: Einordllung der Projekiven und der Konfor- men Auffassung, GOttingen Nachr. 1921, 99-112.

Yano, On complex confurmal connections, Kodai Math. Sem. Rep. 26(1975), 137-151.

Yano, Differential geometry on complex and almost complex spaces, Pergamon Press, 1965-

Yano and S. Bochner, Curvature and Bettinumhers, Ann. of Math. Studies.

32(1953).

K. Y

An invariant of complex conformal connectwns in a Kaehler manifold, Korean Math. Soc. J. 11(1974), 135-139.

A. Hayden, Suhspaces of a space with torsion, Proc. London Math. Soc. 34 (1934), 27-50.

Seoul National University