Some F(3) type structures in the decomposable Riemannian space

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Shonan Institute of Technology

NII-Electronic Library Service ShonanInstitute of Technology

MEMOIRS OF SHeNAN

INSTITUTEOFTECHNOLOCY

VoL 32,No. 1,1998

Some

Ii'(3)

type

structures

in

the

decomposable

Riemannian

space

Irena

CoMi6*

and

Hiroaki

KAwAGucHi**

There are letof books and papers (someof thern are mentioned inthe referencesiFit}) inwhich the

properties of almost complex, or almost product, or _tangent structures were studied separately. Here

they are examined together. Hereare _given some F(3)type structures, definedon the decomposable

Riemannian space, which fordifferentvalues of pararneters become one of the mentioned structures.

AMS Subjectclassification: 53B40. 53C56,53C60.

Key words and _phrases: decomposable Riemannian spaces. F(3)type structures. almost complex structures, almost _product structures, tangent structures.

1. DecomposableRiemannianspaces

Let

(M;

g)bea 4n dimensional C" decomposable Riemannian space. Ifa _pointPEMin some local

chart has coordinates

(x,

_y,u, v),then the allowable coordinate transformations are given by

(1,1)

xa'=xa'(x), _:yi'=yi'cy),uA'=uA'(u), vt'=vl'(v), where we set those ranges of indicesas follows:

a, b,c,d,e,

f=1,

2,...,n,

i,ile,

l,

m, n,_p= n+1,.,,,2n,

A,

B,

C,

D,

EF=2n+1,,,,,

3n,

LJ

K,

L,

A(CIY

P=:

3n+1,...,4n,

a,

B,

r,6,E,rc= 1,2,...,4n.

The

following relations _are valid:

a=i--A=ICmodn),

b=]'=B=1(modn),

(1.2)

c=le=C=K(mod n), d=l=:D=tL(mod n).

We shall suppose that

ranh[AZ'(x)]=ranle[Bl-'(y)=ranle[AA'(u)]=ranle[Bf'<v)]=n, where we apply those notations:

A:'(x)=O.xd(x), Bf･'ly)=Oryi'ly),O.=O/Cixa, Oi--0tbyi

AA'(u)-=O.uA'(u),

Bf'(v)=O,vi'(v).

O.-OIOuA.

0,=0iOvi.

From the above equations

it

follows

that there exist the

inverse

transformations:

(1.3)

xa=xa(x'), yi=[yi(y'),uA=uA(u'), vl=vJ(vr).

Let us denote by

B

the basisof T(M) and

by

B*

the

basis

of 7""(M).We have

B=

{0a}

;

{Oa･

Oi,OA･Oi}

B*=:

{da}

=

{dx

a,

dy

i,

duA,

dvi}

.

* ₁ _VtFfl7

_(--rfx7e7)

_ttfi*l7zz

_Xnv

**

_{fiWIrlt}i:ILiL}

_sw#N

SPdi 9til1O

_n

16 HRN

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NII-Electronic Library Service ShonanInstitute ofTechnology maptIFPJi(\keeeeg 32 g

ee

1e

If

weintroduce thenotations D=AZ'ix)

o

oB

i･

'CY)

o

AA'(u)

OB

R ==

[Cl.

O,Cl,

ooof'(v)Oi],

then the

following

relations are valid:

(1.4)

R'=R'D,

K'=DK.

R' and K' are obtained

from

R and

K

if

in

them the

A ',

I'.D isa regular matrix, so exists DLi, where we

D-'=diag[AgJ(x'),Bl･tv'),AS･

From

(1.3)

it

follows

that

(1.5)

R'=RD-i,

K=p-iK'

The

metric tensor _g

is

determined

by

the matrix

(1.6)

g=diag[A.E1!CL

where those entities are defined by

A=[g.b]=[g

(0.,

Ob)],

E=[gij]=

H-[gAB]-[g(aA,aB)],

1-[gu]-K-duadytduAdvi

indices

a,

i,

A,

construct infact

{u'),Bl(v')].

l

are substituted

by

a',

il

[g

(Oi･

&)]'

[g(&.

ol)].

We

shall suppose that g

is

a symmetric, positively

definite

matrix.

The

tensor gcan be written

in

the form

g=K'gXK

2. Sorne

F(3)

type

structures

Definition 2.1. 71hetensorfietd

F

of

mpe

(1,

1)

dofned

on

M

is

thestnicture

of

F(k)

mpe,

ij'

in

the

basisB itsmatnt can be decomPosed on 4×4 btocles

of

form

nXn, such that ineach row and each

column _are

k

scalar _matn'ces and 4-k _zero btochs.

Notation. Every one of scalar fieldsa, b,_a d denotes the corresponding real or complex

matrix of type n ×n(for example a=a

(x,

M u, v)

L

We shall _givehere sixF(3) type structuresL, i= 1,2,..._,6defined on M; which are fordifferent

values of _parameters almost complex structure

(a.c.s.

J,2･

= -D, or almost product structure

(a.p.s.

Ie=D

or tangent structure

(t.s,

1,2･

=O) .

The common characteristic forallh, iL-1,2,..., 6is

in

the following:theirmatrices expressed

in the basisB of T(ILOhave zero on the place

(1,

1). There are 4!clifferentF(3) type structures

defined

on

M,

concerning the place of zero's.

The structures

Ji,

h

13,

14,

lli,

J6

correspond tothe _{perrnutations} 1234, 1243, 1324, 1342,

1423,

1432

respectively.

The

numbers

denote

therows where the zeros appear, ifthecolumns are _going

innatural succession,

The

complete

list

of

F(3)

type structures will

be

given inthe later.

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Proposition

2.1.

(2.1)

satishes therelations

<2.2)where

a, b,c,

The

indices

he7eand in

(2.3)

Proof.

(2.4)

By directcalculation From

(2.2)

itfollows Proposition 2.2. retations

(2.5)

Proposition 2.3.

(2,6)

satisfies the relation

(2.7)

in

(2,8)

Proposition 2.4.

Some F(3) t](Pestructures inthe decoinPosable Riemannian sPace

7:he

structure

Ji

dofned on M by

d are scalar _.functions

of

(x,

_y,u,

the

fotlowing

satistly therelations

i=a+n,

Te the structure

Ji

1i(0a)=

d(Oi) +e(OA) +f(OD

Ji<OD

==-bcd(O.) -of(OA) +be(OD

jri(OA)=

abe(0.) -of(Oi) -bd(OJ)

Ji(ai)

= _aof(0a) +ae(0i) +cd(0A)

1?=(abe2+aof2-bcd2)L

v) denned on M

A=a+2n,

I=a+3n, a=1,2,...,n.

inthe basisB=

{O.,

Oi,OA,ai}of T(M) corresponds the

J,

=

we obtain

that we findthe several

librbe2+of2 t

(a)

J

{

(b)

J

<c)

J

71hestructure

Jle

dofned on M bythe matn'x

J,=

matrlx O -bcd abe aof

d

O

-of ae e -of O cd '

f

be -bd O

the result that

Ji

expressed by

(2.4)

satisfies

(22),

propositions and a few theorems:

O the structu7e

Ji

inthe basisB determined by

(2.4)

satislies the

?=-I

tff

a=(bcd2-1)(be2+of2)-i

l=

I

_ij7'

a=(bcd2+1)(be2+of2)Li

i=

O

ij7'

a=bcd2(be2+of2)-i.

(empressed

inthe basisB

of

T<mo)

ob-ic-of

ae

bcoabjrabe

-be

of

e

f

c

O

o -c

(c2+2abof)I.

this case,

for

bef7! O we have

(a)

f;

a=(-c2-1)(2bef)-i

(lb,l

li'

g.l(r,s2Eg.)(?-b,fl{)-i

71he

structure

J3

delinedon M by thematrix

(e

4)ressed

inthebasis B)

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NII-Electronic Library Service Shonan 工nstitute of _Teohnology

（

2

．9） 8観頑θs 伽雇 α加 π （

2

．

10

）　　湘南工科大学紀要　第 32 巻　第 1 号　　　　　　0　　 −ab2 　 − b2d　 − b2d 　　　　　　

d

　　 −　

bd

O −

bd

J

，＝　　　　　　a　　　　 O　　　　bd　　　 ab 　　　　　− d 　　 ab 　　 − bd 　　 O 　　　　　　　　／器一（

b2d2

− 2ab2d）1．　　　　ノ彗一イげ a ＝（

b2d2

＋

1

）（

2b2d

）−1 　　　　／§一 ∬ _亘_が α一（b2d2− 1）（2b2d） −1 　　　　ノ莠＝　

O

iff

　a ＝＝

2

−ld ．　　　∬鬼this　case ，ノ

b7

b2d

≠ 0ωe 加 ”召・・11_・

　　

｛

_｛

l

Proposition

2

．

5

．　

The

　s拗 o’z〃召ノ

1 define

（

t

　on　

M

by

　the〃latrix

（2，12）　　　　　　ノん＝ satisfies 伽副覦 oη （

2

．

13

）　　　1鬼thi∫cαsθ，ω ¢ have （2．14＞　　　　　O c−ld （α十わd）召　　　一α〔オθ 　α（α十ウ

d

）召　　　　　　　（のノ

2

・一一1

　

｛

嬲

：

各

乃z _（2．14_）（α）and （2．14）（b）it　is　suPI ）osed 　that　b　

d

十a十abd ≠ Oholds ．

ProPosition

2

。

6

．

The

　structure _ノ』

defined

　on　

M

by

　the吻 α厩κ

（2 ．15＞　　　　　　ノ』＝

satisfies　the　relation

（

2

，

16

）

　　　In　this　case，　we　

have

（2．17）

｛

bce

be

　　　　　　e 　　　ae c−1（α＋わ¢ 尨　　0 　　　0 　　 − （a ＋bd）e 　 − de 　−abce 　　　

O

bde

／

Z

　＝e2（

bZd2

＋α2＋α

bd

）

1

， iff　 e2＝一（b2d2＋a2 ＋abd _）−1 iff　 e2− （b2d2＋α2＋abd _）’i

iff

（e ・＝O_）V _（

b2d2

＋a2 ＋abd ＝

0

_），

　　2　2　　　2 　　　　0 　　　グ 1θ　　うθ 　　　　 θ ad （a十わαのθ　　　

bcde

　　　　　　O　　　　　｛

Z

（α十

bcd

）2 cd _（α十bcd）θ　　　一c2　　　　　aθ　　　　　　　　0 　　−abcde O一αbe−_（a 十bcd_）¢ 　　　　　　　泥＝θ2_（α2＋

b2C2d2

＋abcd ）∬．（α）／§；一∫ 哲7 θ2；一_（α2＋

b2C2d2

＋αうCの一1 ∫彗一1 　げ θ2− （α2＋ b2c2d2＋αうcの一1 （c_{）ノ}

9

＝ O 　 iff _（_θ一_〇_）V _（_α2_＋b2c2d2_＋abcd ＝0_）

In _（2 ．17_）（a_）and _（2 ．17_）（b_）it　is　supPosed ，　that α2＋b2c2b2十abcd ≠ Oholds

　　　Proposition　2。7．　 The　structure ／

6 defined

　on　

M

by

　the　matrix

一 50 一

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NII-Electronic Library Service ShonanInstitute of Technology

Some F(3)typestructures in the decomPosabte Riemannian sPace

(2.18)

k-oaLie

d

c -abc-ad-bc

o

a2d e

o-ac

-be oaribe ad

satishes the relation

(2.19)

JZ=(a2d2-2bce)I.

in

this case,

for

bc l O we have

f

(a)

Jg=-I

iff

e=={a2d2+1)(2bc)J'

(2.20)

t(b)

Jg==I

ij7r

e:-(a2d2-1)(2bc)-i

t(c)

lg=O

iff

e=a2d2(2bc)"i,

Theorem2.1.

Thestructures:

Ji

detennined by

_(2.4)

and

<2.5)

(a),

12

determined

by

(2.6)

and

(2.8)

<a),

h

determined

by

<2.9)

and

(2.11)

(a),

k

determined

by

<2.12)

and

(2.l4>

(a),

Js

determined by

_(2.15)

and

(2.17)

(a),

16

determined by

(2.18}

and

(220}

(a),

are almost comPtex structures

dofned

on

M

If

in

thistheorem sentence

(a)

is

substituted

by

(b)

we obtain almost product structures, and if

(a)

issubstituted by

(c)

we obtain tangent structures, which furnish Theorem 2.2and Theorem 2.3.

3. The tensor character of F(3>type structures

Theorem

3.1.

All

F<3)

tyPestructuresh

i=

1,

2,

...,

6 dojined

on

Mane

tensor

fields

of

mpe

(1,

1)

with respect to the coordinate transhrmations

ofform

(1.1).

The

proof

is

thesame

for

allL

(i=

1,

2,

...,

6)

so thatwe give itforJi. The preciseform of

1i

is

the matrix

J,

= od6

9･e6%.fk59

-

_bcd61'

o7of61 be61 abeac-afoi o-bd6fi aof6Lae6fcd6A o

where the relations

(1.2)

are valid. The tensor

Ji,

which

is

determined

by

the matrix

Ji,

can

be

written inthe

following

way

J,=RJ, XK.

In the basisR' and

K'

Ji

has

the

form

(see

(1.5)):

J,

==R'D-iJ,D X K' ==R7i opK', where

Jl

=D-iJ, D= od6S Bf-･

A

g'

e6%AA･AB'

f69Bf･Ag'

-bcd6{A:･eg1'

o-cfkSl,AA･Bl''

bediBf･Bf-'

abe6:A:･Ag' -of67Bl･v4S' o-bd67BI･AS' acfo'.A:･di' ae6fB;･di' cdGAAA･BI

o

, For1i' we have

1'?=

(DJ

iJ, _D)

(D-iJ,

D)==D-il?D , -51-NII-Electronic Mbrary

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NII-Electronic Library Service ShonanInstitute ofTechnology

uaMIN)<\$me

eg

32

g

ca

1

e

Fromthe above relation itfollows that

ifJ?--I

then

1?--L

ifJ?=I

thenJr'I=L

if1?=O

thenJ'l=O. 1) 2) 3) 4) 5) 6) 7) 8) 9}10) 11) References

Atanasiu Gh,,Natural _pairsof almost complex Finsler structures. The Proc.of fifthNat. Sern.Brasov,

(1988)

67-80.

Bejancu,A. Geometry of CR-Submanifolds,D ReiderPublishingCompany 1986.

Buchner K.Rosca R.Cosymplectic

Quazi-Sasakian

manifolds with aip-structure vector field

e

AnaL StiL

AI.Univ. Al.I.Cuza, IasL 37

₍₁₉₉₁₎

215-223.

Comi6

I.,Niki6

J.,

Some Hermitemetrics inthecornplex Finslerspaces,

{to

be_published inPubl.Inst.Math.

55

(69)

(1994).

Ichijyo,Y.,Almost cornplex structures ef tangentbundlesand Finslermetrics.

J.

Math. Kyoto Univ.6-3

(t967)

419-452.

Kobayashi,Sh.,Nomizu, K.,Feundations of DifferentialGeometry. Interscience Publishers, New York,

London 1963,

Munteanu Gh. Metric almost tangent structures. Anal. StiLAI.Univ, AL LCuza, Iasi33

{1987)

151-165.

Prakash, N. KaehlerianFinslerManifolds.The Math. StudenL Vol,30,No. 1,2,

(1962),

1-11,

Rizza,G.B.StructurediFinslerditipo quasi hermitianoRiv.Mat.Univ.Parma, 4

(1963)

83-106.

Shimada, H.,Rernarkson the almost complex structures of tangent bundles.ResearchReportKushiro

Tech. Coll,No. 21,

_(1987}

169-176.

Yano, K. DifferentialGeometry on Complex and Alrnost Complex Spaces,A Pergamon Press Book, New

York,1965.

'