AMS KEYWORDS.

Internat. J. Math. & Math. Sci.

VOL. 16 NO. 3 (1993) 545-556

545

REAL HYPERSURFACES OF INDEFINITE KAEHLER MANIFOLDS

A. BEJANCUand K.L. DUGGAL Dcpartlncntof Mathematics PolytechnicInstituteofIasi

C.P.17, Iasi 1,6600Iasi Romania

Department

of Mathematics and Statistics University of Windsor

Windsor, Ontario N9B

3P4,

Canada

(Received November 5, 1991 and in revIsed form March 13, 1992)

ABSTRACT. Weshowtheexistenceof

(e)-almost

^contactmetricstructuresand giveexam-

ples of

(e)-Sasakian

^manifolds. Then weget ^aclassification theorem for real hypersurfaces of indefinitecomplex space-formswithparallelstructure vector field.

We

prove that

(e)-Sasakian

redl hypersurfaces ofasemi-Euclidean spaceare either open sets ofthe pseudosphere

(1)

^or of thepscudohyperbolic space

H,_

2aWl

(1).

Finally, weget thecausal character of cosymplecticrealhypersurfaccs of indefinitecomplex space-forms.

KEYWORDS.

(e)-Sasakian

manifolds, ^real hypersurfaces,indefinite complex space-forms, (e)-cosymplecticreal hypersurfaces, globally hyperbolicspace.

1980 AMS SUBJECT CLASSIFICATION CODE: 53C40, 53C50, ^53C55.

0. INTRODUCTION. IndefiniteKahler manifolds have been introducedbyBarros-R.(,mcr

[1].

^Becauseof the signature of themetricwe expectsome essential changesin the study of submanifolds in such spaces. Some new resultson thismatter are obtained in the present paper.

Our

ptu’poseisfirst toinvestigatethe inducedstructuresonrealhyperstu-faces ofanindef- inite Kahler manifoldand thento study some particular classes ofsuch structures. Thus in the first sectionweintroduce

(e)-Sasakian

manifolds which enclose the class of usual Sasakian manifolds.

It

has to benotedthat in thedefinitionofan

(e)-Sasakian

^{manifoldit isessential}

that the causalcharacter of the characteristicvectorfield ofthe structureis preserved.

We

close this section with examples of

(e)-Sasakian

structures on

R

^=’*+1

As

far as we know till now, Takahashi

[9]

^and Duggal

[5]

^{have been} concerned with Sasakian manifolds with indefinite metric.

In

^{section 2 we define an}

(e)-almost

^{contact metric structure on a real}

hypersurface

^of

anindefiniteKahlcr manifold andobtain its principal properties. ^Thenext two sectionsare concerned with two classes of such structures ^on real

hypersurfaces: (e)-Sasakian

^and

(e)-

cosymplectic structures.

In

section 3we show that both the

pseudosphere

2s

(1)

^{and the}

H2n+l

pseudohyperbolic

^{space 2,-1}

(1)

^{areexamplesofspace-like}Sasakian manifolds and time-like Sasakianmanifolds respectively.

Visiting Professor, University of Windsor, Windsor, ^Ontario, Canada N9B ^3P4

1.

(e)-SASAKIAN

MANIFOLDS. Let

M

bea real

(2n + 1)-dimensional

differcntiable lm-mifold endowedwith an allnost contact structure

(f, (,r/).

^{This means that}

f

^{is atensor}

fieldof type

(1,1),

(^isavectorfieldandr/is a1-formon

M

satisfying

Itfollows that

r/of=0;f()=0;

^rank

f=2n. (1.2)

Wethensay that

M

is analmost contact mmtifold

(see

^Blair

[4]).

The manifold

M

is supposed to beparacompact and differentiable of class

O . ^Denote

F(M)

thealgebraof real differentiable functionson

M

and by

F(TM)

^the

F(M)-module

of differentiable vector fieldson

M

Thesamenotation isused for the set ofsectionsofavector bundle over

M

orover any othermadfold.

Throughoutthe paper,byasemi-ILiemannianmetric on

M

weunderstanda,ion-degenerate symmetric tensorfield gof type

(0,2), (cf.

^O’Neill

[8]). ^We

^nowsupposeon

^M

^{thereexists a}

semi-Pdemannian metric_#

(see

^Duggal

[5])

^{thatsatisfies}

g(.f X, .fY) g(X, Y) erl(X)l(Y), VX, Yer(TM) (1.3)

where e +1 It follows that

(X) eg(X,), VXeF(TM) (1.4)

and

e

g(,). (1.,5)

Hence is neveralight-like vector field on

M

This implies that the contact distribution

D {X F(TM), r/(X)= 0}

is always non-degenerateon

M Moreover,

thc indexof g is anodd number v 2r

+

^{1in case is}time-like and an evennumberv 2r otherwisc. This follows as a consequence ofthefact that on

M

wemay consider anorthonormal ficldframe

{E,... ,E,, fE,... ,fE,, }

with

Ei F(D)

^and

such

that

g(Ei,Ei) g(fEi, fEi).

We

are now concerned with theexistenceof semi-Riemannian metricssatisfying

(1.3). ^In

the particularcase e 1 and^r, 0 thereexistsaRiemannianmetricgsatisfying

(1.3)

^and

M

isthe usual almost contactmetricmanifold

(cf.

^Blair

[4]).

^Forthe

^general

^{case,followingBlair}

[4],

^and subjecttothe above mentioned restrictions oftheindexofg we havethefollowing result.

THEOREM ^1. Let

(f,,r/)

^{bean almost} contact structure and

h0

^beasemi-Riemannian metricon

M

suchthat isnot alight-likevector field. Then thereexists on

M

asymmetric tensor fieldg^oftype

(0,2)

satisfying

(1.3)

h0

^wherea

h0 (, )

^and

PROOF.

We

first

^h(Z,Y)

definetwo semi-Riemannian

h(.f2X,.f2Y) + e,(X),(Y),VX,

metrics

hi ^Y - e r(TM).

In

order to prove that his asemi-Riemannianmetric wefirst note that

rl(X) eh(X, )and h(, ,)

^e.

Then denote by

{}

the distribution spanned by on

M

and by

D

^the complementary

REAL HYPERSURFACES OF INDEFINITE KAEHLER MANIFOLDS 547

orthogonal distribution to

{}

^{withrespect to}

h.

^{Then for any}

^X

^E

F(D),

^wehave

,.(x,x) h(-X + ,(X),-X + ,(X)) + ,(X) h(X,X),

since

hl(X,{)

^{0 and}

b.l({,{)

^-e. Thus his asemi-Riemannian metric on

M

ofthe same indexas

h

^on

D1

^{Finally,wedefine} the symmetrictensorfield

(X,r)

1

{h(X,r) + h(lX, It) + ^n(x),(r)}

and wchavc

g(fX, fY)

1

- ^{{h(fX, fY) + h(-X + /(X), -Y + I(Y))}}

g(X, Y) erI(X)rI(Y),

as desired.

Therefore, ingeneral,the above theoren does not provideus aseani-Riexnannian metricon

M satisfying

(1.3). However,

^wemay prove theexistenceofLorentz metricssatisfying

(1.3).

COROLLARY 1. Let

(f,,r/)

^bean ahnost contact structureon M. Then there existsa

Lorcntz metricg on

M

satisfying

(1.3)

^{withe -1.}

PROOF. Since

M

isparacompact there existsa Riemannian metric

ho

^on

^M

^Wedefine b,1,h and_gasinTheorem 1withe -1. Thenit iseasy toseethat bothItandgarcLorcntz metricson

M

Besides,g satisfies

(1.3)

^{withe -1.}

We

call

(f,,,rl,

^g satisfying

(1.1)

^and

(1.3)

^an

(e)-almost

^{contact metric}structure and

M

an

(e)zalmost

^{contact metricmanifold. Thus wehave the}followingnewclasses ofmaxfifolds.

1 e 1 andu 2r.

M

iscalled ^aspace-likealmost contact metricmanifold.

2 e -1 andu 2r

+

^1.

^M

^{iscflleda time-likealmost contact metricmafifol(t.}

An

important subclass of the second classis the

Lorentz

almost contact manifold

(e

-1, u

1),recently

^studied by the second author

(see Duggal [5]). ^As

followingtheterminology of

DuggaJ [5]

and the definition of space-time

(scc

Becm-Ehrlicl

[2])

a timeorientable

Lorentz

almost contact manifold will be calledacontact space-time, ttcrc for the sake ofcompleteness, westate the followingresult

(proved

in

Duggal [5])

^oncontact

space-times.

THEOREM 2.

(Duggal[5]). For

an

(e)-almost

contact metricmanifold

M,

^the following

are equivalent-

(1) ^M

^is contact space-time.

(2)

The characteristic vector field is time-likeand the 2n-dimensional contact distribu- tion

(n,

jr,

g/n)

^isspace-like.

Next,

^{we consider} the fundamental 2-form of the

(e)-almost

^{contact metric} structure definedby

’(X,Y) g(X, fY),VX, Y F(TM) (1.6)

Thenwesay that

(f,,rt,

g isan

(e)-contact

^metric structure ifwe have

(X, Y) dr(X, Y), VX,

Y

F(TM). (1.7)

In

this case

M

is an

(e)-contact

metric manifold. Besides we recall that the almost contact structure

(f,,r/)is

normal if

[f, f] + 2dr/(R)

0,

(1.8)

where

[f,f]

^is the Nijcnhuis tensor field associated to

f. ^An (e)-contact

^{metric structure}

which isnormaliscalledan

(e)-Sasakian

^structure.

A

nanifoldendowedwithan

()-Sasakian

structureiscalledan

()-Sasakian

^manifold.AsinthecaseofRiemamfianSasakian manifolds

wehavc.

THEOREM 3.

An ()-Mmost

contact metricstructure

(f, ^C,

r/,

g)

^is

()-Sasakian

^{if and}only if

(Vxf)Y=g(X,Y)-e7(Y)X, VX, Y r(TM) (1.9)

whereX7 isthe Levi-Civitaconnection withrespect to g Ifwcreplace

Y

by in

(1.9)

^weget

Vx ^{=-eIX, VX} F(TM). (1.10)

Thus,we have:

COROLLARY 2. The characteristic vector field on an

(e)-Sasakian

^manifoldisaKilling vectorfield.

Sasakian manifolds with indefinite metrics have been first considered by Takahashi

[9].

Theirimportance for physics has beenpointedout byoneofthepresent authors

(see

^Duggal Accordingto the causal character of wehave twonew classes of

(e)-Sasakian

^manifolds.

Thus in case is space-like

(e

^{1 and r,}

2r), (resp.

time-like, e -1 mad v 2r

+ 1)

wesay that

M

isa space-like Sasakian manifold

(resp.

time-like Sasakian

manifold}. In

^case

e 1 andv 0 weget thewell-known conceptofRiemannianSasakian mafifold. Ccrt,’finly for physicsit isimportant to consider

Lorentz

metrics.

In

thiscase e -1, ^v 1 andwccall

M

aLorentz-Sasakian manifoldor aSasakian-spacetime

(cf.

Duggal

[5]).

As

Wakahashi

[9]

pointed

out,

fromaspace-likeSasakian structure

(f, ,

r/,g,

e)

^wealways get

a time-like Sasakian structure

(f’, ’, /’, g’, e’),

^where

f’ ^f,

’ ^{-, 1’}

^-/,

^g’

^-g,

^e’

-e and vice versa.

However,

taking into account that the causal character of ^dctcrmiw,s one oranother structureweshallconsider thegeneralcaseof

(e)-Sasakian

structures.

We

close the section with some examples of

(e)-Sasakian

^{structures on}

R2"+x.

Other examplesweshall givein section3.

Firstwe makethefollowingnotations:

Ov,

^the p x k null matrix

I

^{the k x k} unit matrix. Foranynon-negative integer

s

<

nweput

-1 forae

{1,... ,s}

e ^{1 fora}

^{s+l,.

^,n

},

^incases

-

^0,

ande 1 in cases 0

yi z)

1 nascartesiancoordinateson

R

^2"+ and definewith Thenweconsider

(z’,

respecttothe natural field of frames

{ ^-,,

⁰

^--,,

^o

o}

^atensor field

f

^oftype

^(1,1)

^byits matrix.

0,,,, I,, 0,,,

]

[f]=

--[n

On,n 0n,1 01,n eaya

0 Thedifferential 1-form₇is defined by

(1.11)

ifs

#

^0,and

rl dz

+ ^{yidxi- yi*dxi*}

i=1 i*=r+l

(1.12)

REAL HYPERSURFACES OF INDEFINITE KAEHLER MANIFOLDS 549

1=

-

^{dz- .= y’dz ifs=0.}

Thevectorfield is definedfor each s1)y

(1.13)

2ez

0^z

^(1.14)

It is easy to check

(1.1)

^{and thus}

(f,.l)

^{is an} almost contact structure ^on

R

^{2’’+1 for each}

s E

{0,1 n}

^{Finally,we definethe}scmi-Rienaannian metricg by thc matrix

(1.15)

fors

#

^{0, and}

.]

4-

y’y On,, y’

0.,. . ^0.,

y’ 01 ,,,

¹

(1.16)

with respect to the natural fieldof frames.

In

order tohelp thereader tosee the fight form of

[g]

^wewrite it down forn 4 ands 1

"-1

+ ^{(y)2 _yly2} _yy: _yly4

0 0 0 0

y

_yly2

1

+ ^{(y2)2 y2ya y2y4}

^{0 0 0 0}

^_y2

_yya y2y:

1

+ ^{(y3): yay4}

^{0 0 0 0}

^_y3

_yy.i y2y4

yay.i 1

+ ^(y4)2

^{0 0 0 0}

^_y4

0 0 0 0 -1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1 0

y _y _ya _y4

0 0 0 0 1

An

orthonormalfieldof frames withrespect tothesemi-Riemannianmetric

(1.15)

is

0 0 0

yiO

Ei 2--,Ei,

^2--;

^Oy

^i*

^fEi

²

( -Z

f.=2 + ,

Itis easy tocheck that

(f,,/,g)

given by

(1.11)-(1.16)is

^an

(e)-Sasakian

^{structureon/i2n+l}

forany s E

{0,1,... ,n}. In

case s 0 and e 1 weobtainthe classicalSasakian structure

q2n+l on

R

^2’+

(see

^Blair

[4]). ^In

^{othercases wegeteitheraspace-like}Sasakian structureon_"2,

R

^’+

(e

^{-1 s}

# 0) (e

^{1 s}

# 0)

^{or atime-like}Sasakian structureon

2(,-s)+

The Lorentz-Sasakian structureisobtained from the latter fors n.

PHYSICAL

EXAMPLE.

Firstweneed thefollowinginformation

(for

^detailssee

[2,8].

^Let

M

beaspacetimemanifold,withaLorentzmetricg ofsignature

(-, +,... +). A

spacetime

M

iscalledglobally hyperbolicif

M

isaproductmanifoldof the form

(M ^{R S,g}

^-dr

+G)

with

(S, G)

^a compact Riemannian mafifold. Recently the second author, Duggal

[5],

^has provedthe followingphysicalresult,alsovalidfor Sasakian structures.

A. BEJANCU AND K.L. DUGGAL

THEOREM

4

(Duggal [5]). An

odd dinensionalglobally hyperbolic spacetimc^cancarry

tLorcntz-Sasakian structure.

Well known

cxaml)lcs

are Minkowski-spacctixnc,

Lorcntz

spheres ^and Robcrtson-Walkcr Sl)acetimc

[2,8].

In

another direction, physically, Corollary2of Theorem 3isimportant for thespecialcase of Sasakian spacetinessince isaKillingvector field. TheexistenceofKilling^{vectorfieldsin} spacetimes hasoftenbeen usedasthemosteffectivesymmetry.

In

fact,many exact solutions of Einstein field equations have been found by assuming one or more Killing vector fields (Kramer-Stephani-Herlt

[6]).

2.

REAL HYPERSURFACES

OF

INDEFINITE KAHLER MANIFOLDS.

Let hS/

bea real

2(n + 1)-dimensional

^manifold.

Suppose

is endowed with an almost complex structure

.]

andasemi-Riemannian metrict} satisfying

[I(]X, ]Y) [I(X,Y),VX, Y

E

r(T). (2.1)

It follows that the index of

t

^{is an even number t,}

2(r + ^1).

^{Then we say that is an}

indefinite almost Hcrmitian manifold.

Moreover,

ifon wehave

(gxY)Y o,

for any

X,Y r(T.), (2.2)

where

7

is the Lcvi-Civita connection with respect to

.,

^{we say that is aax indefinite}

Kalderianmanifold

(see narros-Romero [1]).

Now suppose

M

is an orientable non-degeneratereal hypersurface

of//

Let N be the normalunitvectorfieldof

M

^Thusby

(2.1)

andtakingaccountoftheorientability ofMwc seethat

-3VN

isavectorfieldtangenttoM. Then the equations ofGaussand Weingartcn

are givenby

xY VxY + h(X,Y)N, VX, Y e r(TM), (2.3)

and

xN ^{=-AX, VX r(TM), (2.4)}

respectively, where X7 is the Levi-Civita connection with respect to the scmi-Rienannim metric ginducedby on

M A

istheshape operatorof

M

and his asymmetric tensorfield oftype

(0,2)

^on

M Suppose

now

[l(g,g)

^{e and}by

(2.1)

^wehave

g(,)

^e. Whc’nfrom

(2.3)

^and

(2.4)

^weget

h(X, Y) eg(AX, Y), VX, Y e r(TM).

Hence

(2.3)

^becomes

TxY 7xY + ^eg(AX, Y)N, VX, ^Y

C

r(TM). (2.5) We

now denote by

{}the

distribution spanned by on

M

and by

D

the complementary orthogonaldistribution to

{}

ⁱⁿ

TM.

Certainly

D

is invariantby

a

and the distribution

{}

iscarried by

a

into the normalbundle. Thus anyrealhypersurfaceofan indefinite Kahler manifold isanexampleofa CR-submanifold

(see

^Bejancu

[31).

The projection morphism of

TM

to

D

isthen denoted by

P

Henceanyvector field

X

on

M

is writtenasfollows

X PX + rl(X), (2.6)

where

r/is

^a1-formon

M

definedby

n(x) (x,o. (2.z)

Thus wehave

r/()

^1.

(2.8)

REAL HYPERSURFACES OF INDEFINITE KAEHLER MANIFOLDS ⁵⁵¹ Further,wedefine a tensor fieldfon

M

by

IX ^]PX,

^VX

^{r(TM). (2.9)}

Then taking account that

D

is invariantby

J

^weget

X ^{-Z + v(Z). (2.10)}

Moreover,

byusing

(2.1), (2.7)

^and

(2.9)

^wc

get

g(.fX,.fY) g(X,Y)-ev(X)(Y),VX, Y r(TM). ^(9..11) Hence,

^wcobtain

PROPOSITION 1.

An

oricntablenon-degeneratereal

hypersurfacc

^ofanindefinitealmost Hcrmitian manifoldofindexu 2r inheritsan

(e)-almost

^{contactxnctricstructure}

(.f, f,

q,

g).

Moreover,

wehave

PROPOSITION 2. The

(e)-almost

^{contactlnctricstructureon}

^M

^{imncrscdinanindefinite}

Kahlcrian manifold satisfies

and

(Vxf)Y q(Y)AX eg(AX, Y),

(Vxy)Y eg(.fAX, Y), (2.13)

COROLLARY3. Let

M

beasinProposition2. Then thefollowingassertions^areeqlfiwtl.nt

(i) f

^isparallelon

^M

(ii) /is

^parallelon

M (iii) f

^isparallelon

^M

(iv)

^Theshape operatorsatisfies

AX (AX)f, VX F(TM). (2.15)

We

nowrecall fromgeneral theoryofhypersurfacesin semi-Riemannianmanifolds that the Gaussand Codazzi equationsaregiven by

g((X,Y)Z, W) g(R(X,Y)Z, W) + ^{g(AX, Z)g(AY, W)}

(2.16) g(AY, Z)g(AX, W),

and

9((X, Y)Z,N) 9(7xA)Y ^-(VyA)X, Z), (2.17)

respectively, for any

X, Y, Z, W

E

F(TM),

^where

/

and

R

are the curvature tensor fields of and

M

respectively.

On

the other

hand,

^we recall

(see Barros-Romero [1])

^{that the}

curvaturetensorfieldofanindefixfiteconplex-spaceform

)Q(c)

isgiven by

k(X,Y)Z {(Y,Z)X ^{[I(X,Z)Y + [I(Y,Z)X} .5(X,Z)Y+}

+ 2(X, YY)YZ }

(2.18)

foany

X, Y, Z e r(T).

for any

X,Y F(TM).

PROOF.

By

direct calculations in

(2.2)

^using

(2.4)

^and

(2.5)

^{we obtain}

(2.12)

^and

(2.13).

Thenwe replace

Y

in

(2.12)

^{by andobtain}

(2.14).

From

Proposition2we easilyobtain

.fAX, (2.14)

Thenwehave

THEOREM 5. Let

M

be a connected real hypersurfacc with dimM

>

3 ofan indefinite complex

space-form/(c)

satisfyingone oftheassertions ofCorollary 3. Then c 0 ,’rod

M

is ascnfi-Euclidcaa space.

PROOF. We replace

Z

by

PZ

in

(2.17)

^{andby using}

(2.15)

^{and assertion}

(iii)

of Corollary 3,wcobtain

g((X,Y)PZ, _N)

0. Then from

(2.18),

takingaccount of

(2.6)

^wc get

(Pz,]g)O(x,()-(Pz,]x)(Y,() o,x,g,z e r(TM)

whichimplies

ce

(2.19)

-( ^{]PZ, Y)}

⁼⁰

Suppose

now c

:/:

^{0 and from}

(2.19)

^weget

PZ

0 for any

Z

E

F(TM),

^whichcontradicts the hypothesis dimM

>

3. Thus c 0 and byusing

(2.15)

ⁱⁿ

(2.16)

^weobtain

^R

^{0 which}

completesthe proof.

3.

(e)-SASAKIAN REAL

HYPERSURFACES OF AN

INDEFINITE KAHLER

MANIFOLD.

Firstweobtain thefollowingtheorem ofcharacterizationfor

(e)-Sasakian

^realhypersurfaccs of indefinite Kahler manifolds.

THEOREM 6. Let

M

bean orientable real hypersurfaceofaaindefinite Kahlcr manifold

1/.

Then the following assertions with respect to the

(e)-almost

^{contact metric structure}

inheritedby

M

areequivalent"

(i) M

^isan

(e)-Sasakian

manifold,

(ii)

^The

(e)-characteristic

vector field satisfies

(1.10).

(iii)

^Theshape operatorsatisfies

AX

-eX

+ ( + ,(A))(X),

VX

F(TM). (3.1)

PROOF.

(i) = ⁽ⁱⁱ⁾

^{was shown}in section 1.

(ii) (iii). By

using

(1.10)

^and

(2.14)

^weget

PAX -ePX, VX F(TM).

Hence by

(2.6)

^wehave

AX

-ePX

+ ^{I(AX), X F(TM).}

From (3.2)

follows

A ^{/(A). (3.3)}

Finally,taking account that

A

isa symmetricoperatorwithrespect to g andby using

(1.4), (2.6)

^and

(3.3)in (3.2)

^weobtain

(3.1).

(iii) (i).

^Replace

^AX

^from

(3.1)in (2.12)

andobtain

(1.9).

In

order to state the next result werecall

(see

^O’Neill

[8]),

the definitions ofhyperspheres and pseudohyperbolic spaces in semi-Euclidean spaces. Consider the semi-Euclidean space

R("+x)

_2s with theindefiniteKahlerianstructure

(cf. Barros-Romero [1]).

^Thepseudosphereof radius r

>

0 in_"2 isthe hyperquadric

-{

^{r2("+i) =r}

2}

REAL HYPERSURFACES OF INDEFINITE KAEHLER MANIFOLDS 553 ofdimcnsion

(2n + 1)

^{andindex 2s.}

In

a sinfilar way, the pseudohypcrbolic space of radius

2(n+1)

v

>

^{0 in}

R2s

^isthe hyperquadric

2,+l

{

^2(n+l)

_r2}

ofdimension

(2n + ¹⁾

^andindex

^{(2s 1).}

Wenow state

THEOREM T. Let

M

bean

()-Sasakiaai

connected realhypersurfaceofo2("+1)

.

^Then

^M

isanopen set eitherof

"-’2sq2n+l (1)

^{orof HZn+l}"’28--1

(1).

2(n+1)

PROOF. Since

R2,

^isafiat indefixfitecomplexspaceform,from

(2.17)

^weobtain

(VxA) ^Y (VrA)X

0,

VX, Y e F(TM) (3.4)

Next,

from

(3.1),

^weget

AX -eX, VX r(D),

^and

(3.3). (3.5)

The,, wetake

X

E

F(D)

and

Y

in

(3.4)

and by using

(3.5), (3.3)

and takingintoaccount that

Vx

^and

^{V qX belong}

^{to the}contact distributions

D

weobtain

r/(A)

^-e

(3.6)

Hence byusing

(3.6)in (3.1)

^weget

AX -X, VX r(TM). (3.7)

ThereforeMis atotally umbilicalhypersurface

(but

^nottotallygeodesic)withnormalcurva-

turek -e

Hence

byLelnma 35 andProposition36fromO’Neill

[8],

^{p.l16,wcobtainthat}

M

has constant curvature eandit isanopen set of_’28

(I)

^{when e I and an}open set of

z,_a

(1)

^whene -1.

Suppose

now

M

is a totallyumbilical real hypcrsurfaceof

.,/,

that is,

A pI,

where p is adifferentiablefunctionand

I

is the identityon

F(TM).

Thenwefirst state

(,,+x)

THEOREM 8.

A

real hypersurface of

Rz,

^is

(e)-Sasakian

if and only if^it is totMly umbilical and p -e.

PROOF. The firstpart ofthe assertionfollows fromtheproof of Theorem 7.

Suppose

now

M

is totally umbilical with p ^-e. Then

A -e

^{and thus}

^r/(A)

^-,.

^{Hence (3.1)}

^is

satisfied and tiffscompletestheproof.

REMARK

1. Tashiro

[10]

hasconstructed theSasakianstructureon asphereofaEuclidean spaceandTakahashi

[9],

byadifferent

approach

thanours, obtained the

(e)-Sasakian

^structure

q,2n+l

H2,_1

2n+l

(1) on,:s (1)

^and

Now

suppose

M

isatotallyumbilicalreal

hypersurface

ofanindefinitecomplexspaceform

//(c).

Thenweget

g

((VxA)Y ^(VyA)X,) O, VX, Y r(D). (3.8)

On

the other

hand,

from

(2.18)

^weget

9((X,Y),N)

c

- ^{g(X, fY), VX, Y F(D). (3.9)}

Hence

from

(3.8)

^and

(3.9),

takingaccountof

(2.17) we.obtain

^{c 0,whichenableustostate}

PROPOSITION 3. There exist no totally umbilical real hypersurfaces in an indefinite complexspaceform of non-null holomorphic sectionalcurvature.

Tashiro-Tachibana

[11]

first obtained sucharesultforpositive^definitecomplexspace forms.

4. COSYMPLECTIC

REAL

HYPERSURFACES OF

INDEFINITE KAHLER

MANIFOLDS.

Suppose

as in the previous section

M

is an orientable real

hypersurface

^{of an indefi-}

nite

2(n + 1)-dimensional

Kalflcr manifold

b;/.

_{Then we} say that the

(e)-almost

^contact

metric structure

(f, ,

^/,

^g)

^{induced on}

^M

^{defines an} (e)-cosymplectic structure if both the 1-form ₇ and the flmdamcntM 2-form given by

(1.6)

^{are closed.}

^M

^is then callcd an

(e)-cosymplcctic hypersurface. Thereforeon

M

wehave

&I(X, Y)

0, and

(4.1)

d@(X, Y, Z)

0, ^forany

X, Y, Z F(TM). (4.2)

Ifwccxpress

(4.2)

^bymeansof the Levi-Civitaconnectionweobtain

d(X,Y,Z)

1

- {9(X,(VzflY) + 9(Y,(Vx.f)Z) + 9(Z,(Vrf)X)} (4.3/

Then using

(2.12)

and

(2.7)

ⁱⁿ

(4.3)

bydirectcalculationsitfollows that

(4.2)

^{isalwayssatisfied}

on

M

Hence

M

isan

(e)-cosymplectic

manifold if andonlyif

(4.1)

^issatisfied. Furthermore PROPOSITION4.

M

isan

(e)-eosymplectie

hypersurfaeeif andonlyif theshape operator atisfies

Ao]’+foA=0. (4.4)

Theprooffollows from

(4.1)

taking^intoaccount that

&I(X,Y)

1

- {(Vx,I)Y- (Vr/)X}, VX, ^Y F(TM)

and by using

(2.11)

^and

(2.13).

Fromthispropositionweinfer

COROLLARY 4. Let

M

bean (e)-cosymplcctic ^real hypersurface ofanindcfinitc Kahlcr

amnifold//

Then wehave

(i)

^isaprincipal curvature vectorfield,

(ii)

the trajectoriesof aregeodesics.

PROOF. Apply

(4.4)

to and obtain

PA

^0.

^Hence

^by

^(2.6)

^we

^get

A a,

a

/(A) (4.5)

whichmeans that is aprincipal curvature vector. The secondassertion follows from

(4.4)

by using

(2.14).

REMARK

2.

(4.5)

follows fi’om

(3.1). ^Hence

^{the firstassertionofCorollary}4 also holds for

(e)-Sasakian

^realhypersurfaces.

With respect totheexistenceof(e)-cosymplectic^realhypersurfacesimmersedincomplex space forms such that theirshape operatorshavereal eigenvalues, ^weobtain

THEOREM 9. Let

M

be an (e)-cosymplectic real hypersurface ofan indefinite complex

2(n+l

spaceform

M, )(c)

such that theshape operator

A

hasonlyrealeigenvalues. Then

(11

Ifc 0, then

M

isasemi-Euclidean space.

(2)

Ifc

t

^{0, thenwehave}

(a)

^c 4, and

M

should betime-like

(b)

^c -4,and

M

shouldbespace-like

Moreover,

inthe last two cases,

M

has at most threeprincipalcurvatures.

REAL HYPERSURFACES ^OF INDEFINITE KAEHLER MANIFOLDS 555 PROOF.

By

direct calculations taking^accountof

(1.5), (2.13), (2.14)

^and

(4.5),

^weget

g(,,(VxA)Y) +

^g(Af

^{AX, Y)} eX(ot)l(Y) + 2g(f AX, Y), (4.6)

for_any

X, Y

E

F(TM).

^{On theother hand,}from Codazzi equation

(2.17)

^{taking account of}

(2.18)

^weobtain

C

"X

g((VxA)Y-

(VrA)X,f)= g( ^{,/Y) VX, Y e F(TM) (4.7)}

Then takingaccount of

(4.4)

^{wc sccthat}

(4.6)

^and

(4.7)

^inply

g(X,

.fY

C

^{2A]:AY) e{X(a)t(Y) Y(a)y(X)}. (4.8)}

Takenow

X

in

(4.8)

^adobtain

r(a) f(a)(Y), ^VY r(TM), (4.9)

which togetherwith

(4.4)

^and

(4.8)

imply

Cy + A= ^Y

^0,

^VY r(D). (4.0)

4

As wc have seen in Corollary 4, is aprincipalcurvaturevectorfield of

M Suppose

now

Z F(TM)

^{is anotherprincipal} curvature vector field of

M

and

A

^q

R

isthe corresponding p.rincipal curvature. Thenbyusing

(2.6)

^and

(4.5)

^weget

APZ- APZ + ^{rt(Z)(a- )}

^0.

^(4.11)

But

taking account that

A

is a symmetricoperator with respect to g and using again

(4.5)

weobtain

g(APZ,,)

g(PZ,

A,) ag(PZ,,)

O, which togetherwith

(4.11)

^imphes

APZ=APZ. (4.12)

We

now replace

Y

from

(4.10)

^by

^PZ

^andobtain

ce₄

+ A2

^0o

(4.13)

In

case c 0 we then have 0 and thus

AY

0 for each

Y F(D)

^{since the} cigcn distributionof

A

withrespectto thiseigenvalueisjust D. Further, byusing

(2.6), (2.7)

^and

(4.5)

^weobtain

g(AX, Z) ea,(X)(g), VX, Z e r(TM). (4.4)

Thentaking ^account of

(4.14)in (2.16)

^weinfer

R(X,Y)Z

^{0 for any}

^{X,Y, Z} . ^r(TM).

Hencewehavetheassertion1 of the theorem. Theassertion2 follows from

(4.13)

takinginto account thatthecigenvaluesof

A

aresupposed to bereal.

COROLLARY

5.

Let M

be eitheraspace-like cosymplecticreal hypersurfaceofrmindefi- nitecomplex space-formof positiveholomorphicsectionalcurvatureoratime-likecosymplec- tic real hypersurface ofan indefinite complex space-formofnegative holomorphic sectional curvature. Thentheshape operatorof

M

has atleast twoeigenvalueswhicharenotreal.

REMARK

3.

In

thecaseofcosymplecticrealhypersurfacesof positive definite space

forms,

important resultshave been obtainedbyOkumura

[7].

Next

by

(4.1)

^{we see} that the distribution

D

is involutive on an

(e)-cosymplectic

real hypersurface

M Moreover,

in case 2 ofTheorem 9 by using

(4.4)

^{we derive} that

A

has eigenvalues

(+1)

^and

(-1)

^{with the same}multiplicity n.

Denote

by

D

⁺ and

D-

the eigen distributions withrespect totheabove eigenvalues. Further, take

X, Y

q

F(D +), ^Z

^q

F(D)

andfrom

(2.17)

weget

g

(IX, Y] A([X, Y]), Z)

^O.

On theotherhand,byusing

(4.5)

^{and takinginto accountthat}

^D

^isinvolutive,^weobtain g

(IX, Y] A([X, Y]), ’)

^O.

Hence

A([A,X]) [X, Y],

which says that

D

⁺ isinvolutivc.

In

asimilar way, itfollows that

D-

isinvolutive too.

Suppose

nowthat

M

⁺isaleaf of

D

⁺ and denote h

+

_and

.+

the second fimdamental forms ofinnncrsionsof

M

⁺ in

M and/t/(c)

respectively. Then forany

X, Y

E

F(TM +)

^wchave

xY V.Y ^{+ h+(X,Y) + eg(X,Y)N,}

..Y V.Y + ]t+(X,Y),

where X7+is the Levi-Civitaconnectionon

M +.

Thuswehave

PROPOSITION 5. Let

M

⁺ be aleaf of

D

⁺ which is totally geodesic immersed in

M

Then

M

⁺ istotally umbilical immersedin

)t/(c).

Certainly sucharesultholdsfor leavesof

D-too.

ACKNOWLEDGEMENTS.

We wish to thank the

NSERC

^of Canada and the Research Board of the Ufiversity of Windsor forsupportingthis researchwith the award ofresearchgrants.

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