Hypersurfaces in a conformally flat space with curvature collineation

VOL. 14 NO. 3 (1991) 595-604

HYPERSURFACES

IN

A CONFORMALLY

FLAT SPACE

WITH

CURVATURE COLLINEATION

K.L. DUGGALandR. SHARMA

Department

ofMathematicsand Statistics

UniversityofWindsor

Windsor, Ontario, CanadaNgB _3P4

Department

ofMathematics

University ofNew Haven West

Haven,

Connecticut, 0fi51fi,U.S.A.

(Received February 7, 1990 and in revised form August 13, 1990)

ABSTRACT.

Weclassify theshapeoperators ofEinsteinandpseudoEinsteinhypersurfacesina

conformallyflat space withasymmetry called curvaturecollineation. Wesolvethefundamental problemoffindingall possibleformsof non-diagonalizable shape operators. A physical example of space-timewithmatterispresentedtoshow that the energyconditionhas direct relation with

thediagonalizabilityofshape operator.

KEY

WORDS AND PHRASES: Shapeoperators,

Proper

and improperHypersurfaces,Einstein

and pseudo-Einsteinhypersurfaces,curvature collineation, mattertensor.

1980AMSSUBJECT CLASSIFICATION

CODE:

AMSSubjectclassificationcode: 53(340,53C50

1.

INTRODUCTION.

The eigenvectors of theshapeoperator

(second

fundamentalform

opera-tor)

ofahypersurfaceofasemi-Pdemannian spaceneed not be all real. Theyarerealfor positive

definitehypersurfaces. Forindefinitehypersurfacessomeof them may becomplexandzerolength

(null).

In

thelattercasethe real eigenvectors (principal

directions)

maynot span the tangent space of thehypersurface at every point. If the eigenvalues (principal normal

curvatures)

are

real and noeigenvectorsare null at every point, then the hypersurfaceis called proper in the terminology usedbyFialkow

[1].

A

hypersurfacewhichisnot proper,iscalled improper.

Usually oneprefers to embeda hypersurfacein afiat

(Euclidean

or

Minkowskian)

space.

But,

aspointedoutby

Goenner

[2],

thereisnospecificreasonforchoosingthe ambient spacefiat, one canconsiderother ambientspaces suchas aspace

form,

aRicci-flat space,anEinsteinspace and aconformally fiat space. Fialkow

[1]

provideda completeclassification ofproper Einstein

hypersurfacesinanindefinitespaceform. Magid

[8]

hasalgebraicallyclassified improperEinstein

Theaimofthis paperis topresent anMgebraicclassification ofproper/improper Einstein

andpseudoEinsteinhypersurfacesin aconformMlyflatspace.

An

exmnpleofthisspace-timeis consideredto show that the energyconditionhasdirectrelationwith thediagonalizabilityof the shape operator.

DEFINITION

1.1.

(Katzin

et al

[4]).

A

vector field Vin asemi-Pdemannianspaceissaid togeneratea1-parameter group of curvature collineations ifitsatisfies:

LvR

0, where

Lv

and

R

denote theLie-derivativeoperatoralong Vand theRiemanncurvaturetensor, respectively. Curvaturecollineation

(6"(7)

is afundamental symmetry

[4]

property ofsemi-Pdemannian spaces. Indeed, it is known

[4]

that, for Ricci-flat spaces, more familiar symmetries such as

projective and conformal collineations(includingaffinecollineations,motions,conformaland

ho-mothetic

motions)

aresubcasesof (76’. Physically, the well-knownKomar’scovariantidentity

[8]

(which

acts as aconservationlawgeneratoringeneral relativity)follows naturallyasanecessary condition for a 6’(7. Thus C’(7’s arenecessarily embraced by the group of

general

curvilinear coordinatetransformations.

We, therefore,

chooseaconformallyflatembedding

space/r

admittinga1-parametergroup of (7(7’s. Our choice of the embedded space is a class of semi-Riemannian spaces defined as

follows:

DEFINITION

1.2.

A

semi-Riemmmian space is saidtobe pseudo-Einsteinifthereexistsa

1-formusuchthat

Ric xg

+

flu(R) u

(1.1)

and

g(U,U)

e(e

1);

where

g(U,X)=

u(X),

Ric denotes theRicci tensorandx,r/are scalar functions.

For

r/=0,itreduces to anEinstein space.

Theabove definitionismotivated by

(1)

Yano’sdefinition

[11]

ofapseudo-EinsteinhypersurfaceofaKaehlerian space, givenby

equa-tion

(1.1)

withX, r/asconstmats.

(2)

Einstein’s fieldequationsin theframework of

general

relativity

[4]

me+{^-

}

I

+

(

+

)

(R)

,

wherep

and/z

rethe pressure and energy densities ofaperfect fluid,tisthe1-formmetrically equivalent to the velocity 4-vector

U

and A stands for thecosmologicalconstant.

2.

SHAPE

OPERATOR

OF

EINSTEIN

HYPERSURFACES.

Firstwestate thefollowingresult ofKatzinet al

[5].

LEMMA

2.1. Let avectorfield

V

generate 6"6"in anm-dimensionalconformM1yflat space

.

with metric g. If

r

is a space form then

V

defines amotion

(isometrr)

for m

_>

3 nda conformM motionform 2. If

r

isnot&spaceformthen

Lvg

2ag

+

TRi---’,

whereo"and

arescalr functions and

Rc

isthe Pdccitensor ofir.

(scMar

curvature

#

0)

themetric is

/to

()

e(),

d

2=ex(dz1)+

1-where

e’s

e l,K0 is the constt curvature of K_ d upper ee inces run from 2 to n.

An

example of such spaces is the Einstein Static

Uverse.

Moreover,

the

CC

vector is

V

[V(z1),

Va(z,...

,z)]

where V is bitry function ofz other th az1

+

bz

+

e

and Va defines a motion inK,_. Thismes that K_ is atoffy geodesic hypersurfaceof

K1

xK_. Thuswe are motivated tosumethat the tgentiMcomponent ofthe CCvector

in generatesamotion inM.

As theeewhen is spaceform,hbeenseussedin

[9],

weshM1assume not to be a spaceformin the subsequentdiscussions.

TEOREM

2.1. IfanindefiniteEinsteinhypersurface

M

ofdimensionn 3,isembedded isometricMly intoa conformy flat space adtting

GG

generated by avector fieldwhose tangentiM component isKilling in

M,

then the shape operator

A

at each point of

M

is either

diagonMizableor cbe putintooneof thefollongforms:

A

a 1 0 a 1

0 a

a b

whereaand barearbitraryfunctionson

M,

withrespecttosomespeciMlychosenbasis. Inthe

last case,nisevenand the signature of themetricof

M

is

(n/2,n/2).

PROOV:

A

striaghtforwardapplication of lemma 3.1 shows

Lvg

2og

+ TI

(2.3)

where

V

isthe vector field generatingCCin

.

We

decompose

V

int

itstangential part and normalpart

N

as:

V

+N,

(2.4)

N

denotingtheunitvectorfield

(g(N,N)

+1)

normalto

M

and

f

ascalarfunction onM. Letus denotearbitraryvector fields

tangent

to

M

by

X, Y, Z,

W.

Theneqn.

(2.3)

obtains

g(xV,

Y)

+

g(yV,

X)

2og(X,Y)

+

T--(X,

Y),

where

7

denotes Levi-Civita connection ofJ/. Employingeqn.

(2.4),

Gauss and Weingarten formulaeinthe above yields

As

isconformallyflat, wehave

g((t,

?)2,

)

[--(,

)g(?,

)

--(t,

2)g(?,

)

+

g(?, 2)Ri(X, if’)

g(X,

)mc(Y,

)]/(.

Z)

{/n(n-

1)}[g(]P,)g(’,)g(,])]

(2.6)

TheGaussequationfor

M

is

9(.(X,

Y)Z, W)

9(R(X,

Y)Z, W)

e{g(AY,

Z)g(AX,

W)

g(AX,

Z)g(AY,

W)}

(2.7)

where

A

istheshape operatorofM. Usingit in

(2.6)

gives

g(R(X,Y)Z,

W)

[Ric(Y,

Z)g(X,

W)

rtic(X,A)g(Y,

W)

+

g(Y, Z)Ric(X,

W)

g(X,

Z)mc(r,

w)]/(n

1)

(2.8)

{e/n(n- 1)}[9(Y,

Z)g(X, W)-

g(X,

Z)g(r,

W)]

+

[a(AY, Z)a(AX, W)

a(AX, Z)(AY,

W)]

Now,

ifwesubstitute)

lTd

N

and

17"

Y,

2

zin

(2.6),

then

g((N,Y)Z,N)

[{Ric(N,N)

(ee/n)}/(n

)]g(Y, Z)

+

(/(.

1)Pdc(Y, Z).

By

the substitution

X

W e; where

{e}

is an orthonormal frame in

M

in eqn.

(2.7),

multiplying byei

9(ei, ei)

and finally summingoveri,wefind

Ri---(Y,

Z)

eg(t(N,

Y)Z,N)

Ri---(Y,

Z)

e{tr.A)g(AY,

Z)

g(AY,

AZ)},

keepinginmindthat

{

ei,

N}

constitutesanorthonormalframein/r. Eliminating

g((N,

Y)Z, N)

from thelast twoeqns. gets

rue(Y, z)

{(n- 1)/(n- 2)}[mc(Y,Z)

+

{(enrti--7(N,

N)

e)/n(n

1)}g(y,

z)

(2.9)

+

e{(tr.A)g(AY,

Z)

g(AY,

AZ)}]

Again, substituting

Y

g _{el,multiplyingeiand then summing}overi;intheabove eqn. yields

-

r

2eRi--(g,N)

e{(tr.A)

tr.A}.

(2.10)

By

feedingeqn.

(2.10)into

(2.9)

weobtain

Rie(Y, g)

{(n

)/(n

2)} [Rio(Y, Z)

e{(tr.A)g(AY,

Z)

g(AY,

AZ)}]

g(X,

Y)[(n

2)(F/n)

,"

+

e((/r.A)

tr.A’)]

Througheqn.

(2.5)

and thehypothesisthat

generates

motionin

M,

wefind

Ric(Y,

Z)

-(2.f

/r)g(AY,

Z)

+

pg(Y,

Z)

where pis a scalar function on

M. A

cumbersomecomputation using the last two equations shows

Q

I

+

OA

A

Q

beingdefinedby

g(QX,

Y)= Pat(X, r),

and

enb

r

+

{2f(n

2)/(n

1)}tr.A

e[(tr.A)

2

tr.A

}

(2.11)

8 7tr.A

2ef(n

2)/(n

1)

(2.13)

Now,

using the Petrovclassificationscheme

[10]

for symmetrictensors,

A

can be cast intothe form:

where

isan

(i

x

i)-matrix

and

nl

C1

di,i

di

0 di,,i

di

Bi

"’.

(di

+l)

"’.

di

di,,i

isa

(2tj

x

2tj)-matrix. As

perourhypothesis,

M

isEinstein,i.e.

O

(r/n)I.

Thus,eqn.

(2.11)

implies that the orders of blockmatrices

Bi

and_Cj are

<_

2. Consequently,

A

has blocks oftype:

dAj

dj

o

[,]o,

-ortheircombination. Thesecond blockcanbe transformedinto

x,0

x,1

by change ofthe basis:

{e,

’}

{-e,

’}.

Eventually,eqn.

(2.11)

canbeputas

(er/n)I

I +

0

M’atching

correspondingentriesweobserve

0 2Aj,

(0

2ak)lk

0

(2.14)

where

(er/n)

-$. If thereareanyblocks

with/l’s (/5’s

being

non-zero)

then a Aj

19/2,

foreveryj and k.

Hence

a’s

and

A’s

areallequalto

0/2.

From

the last relation in

(2.14)

wealso observethat

fl’s

are rdlequal. Thus therelation-set

(2.14)

isequivalentto:

0=2A,

0=2c,

vi

(19

+ V/(192

40))/2,

Obviously,

A

cannot haveboth a and

A

blocks,otherwise

A

a

and/5t

0.

So,

if

A

has only A-blocks thenvi

A

and

A=

0

A

1

0

A

tr.A

nA

A

1

0

A

2ef/(n-

1)T.

Consequently,

A

a 1

0 a 1

0 a

"..

a 1 0 a

where

a’r(t- 1)

2el.

On

the other

hand,

if

A

has only a-blocks then

v

a4-

V/(1

-2)

whichcannot bereal,because

B

#

0;and hence there cannotoccurany v-term inA. Eventually, wefindthat all

a’s

0/2

and

A=

Thus,0 2a aand

"-.

tr.A=na=n0/2.

a b

Thiscompletes theproof.

COROLLARY

1. Underthe hypothesis of theorem 2.1,if

M

isimproperthen, either

(A

aI)

2

O,

_or

(A- aI)

-b:I,

_{a, bbeing arbitrary functionson}

M

_and

I

_{the identityoperator.}

3.

SHAPE

OPERATORS OF PSEUDO-EINSTEIN HYPERSURFACES. In

this section we studypseudo-Einsteinhypersurfacesin thesamecontextasof the

pre,ceding

section. Weprove thefollowingtheorem for 4-dimensionalhypersurfaces only becauseit iscumbersome to consider theblocks of theshapeoperatorinhigherdimensions.

THEOREM3.1. Ifa4-dimensional pseudo-Einsteinhypersurface

M

4_{isisometrically}

embed-dedintoaconformallyfiat space jr5 admitting

CC

generated byavector field whosetangential component generatesmotionin

M

4,

thentheshapeoperator

A

of

M

4 _{iseitherdiagonalizableat}

each pointor canbeputintooneofthefollowingforms:

a a b

A=

a or -b a or

a 1 a b

Oa -ha

wherea,b,v andr/are scalarfunctionon

M

4 _{and e e}2 _{1, with respect to somespecially}

chosenbasis.

PROOF: Withrespecttoanorthonormalbasisformedby

U

andthreeorthonormalvectors orthogonalto

U,

weobserve that theRiccimapcanberepresented by

z+er/

wheree

g(U,

U)

+1. Proceeding exactly asin theproofoftheorem 2.1, upto eqn.

(2.11)

we

findthat

A

hasblocks of type

[]or

0

i

or

_

=

Therefore,either

A

willbe diagonalizableateach point orcanbe putinto oneof thefollowing forms:

(1)

v’

A

1

(2)

A1

1

0

’1

(4)

(3)

A2

1

0 A,

Plugging the above listed forms of

A

intoeqn.

(2.11)

shows that the type

(2)

is not possible whereas thetypes

(3)

and

(4)

hold only if

M

4_{isEinsteinwhichistakencareofbytheorem2.1.}

Theremaining type

(1)

leadsto:

Thus,in thiscase

A

v 1

0 v

Wherev

0/2

and

tr.A

(8ef/3T)

:

V/(--ee/).

COROLLARY

2.. Under the hypothesis of theorem 3.1, if

M

is not Einstein then

A

is eitherdiagonalizableateachpointor canbeputinthe form

A

v 1

withrespect tosomespecially chosenbasis.

We

now interpret the physical significance ofa pseudo-Einstein space in the light of the abovecorollary. Einstein’sfieldequationsofgeneralrelativitycanbewritten insuitableunitsas:

1

Pdc

+(^-

-r)g-

T

(3.1)

whereTis theenergy-momentumtensorofmatterdistribution.

A

comparisonof

eqn.(3.1)

with

thedefining

eqn.(1.1)

ofapseudoEinstein space,showsthat

T

p

+

(p

+p)u

(R) u

ateach pointor

a

Onthe otherhand,if the signature ofjQ5 is

0 v

-+ +

+,

thenwehave6 e -1,

V/(-e)

V/{-(p

+

p)}

which is always imaginary for physically realistic matter. Consequently,

M

is proper. For thislatercase,the energy condition: p

+

p

>

0,is equivalenttosaying that

M

4 _is

proper.

4.

EXAMPLE.

Isometricembeddingsof space-times has been used to obtaindeeper insightinto

thegeometricalproperties of the embedded space-time. Thistechnique has also shedsomelight

onanumberofglobalquestionsconcerningsingularitiesand causality properties.

In

particular, severalphysicallyuseful exact solutionshave been found bytheembeddingtechnique, at least forsome casesof low embedding class

[the

maximumnumber of extra dimensions is called the embedding

class].

In

fact,the maximalanalyticextensionof the Schwarzschildsolution

[3]

was

foundbythe method of embedding.

In

supportof theabove,we present anexampleofanisometric embeddingof the typeof space-time described in this paperviz., 4-dimensional space-time of

general

relativitywith an

isotropic matter oftype

I

[3].

For details,werefer

[71.

A

4-dimensionalspace-timeis ofembedding class 1

(i.e.,

a

hypersurface)

ifthereexists a

symmetric tensor

fb

satisfying:

]abcd

e(’acbd

’ad’bc),

e 4-1

(Gauss)

fb;

fc;b

(Codazzi)

wherepand_{# are}givenby

1

p+-r-^=X,

p+p=

r/

(3.3)

Eqn.

(3.2)

representsanisotropic matter oftype

[3]

withenergy densitypand pressure p, provided

g(U,

U)

-1 (signatureof

M

being

+

++).

It

isremarkable to observe that sucha space is non-Einsteinbecauseotherwisewewouldget the non-physical state

+

p 0.

Thesignatureof

hr5

couldbe

-++++

or

--+++

only. If

M

has the former

(Lorentzian)

Thefieldequations then yield

(for

A

0).

1

Rab

Tab

+

rgab

e(

abCc

ae’

All possible tensors f correspondingto the isotropic matter or Maxwell typeare known. Precisely, therearefourdifferent cases

(for

details,see

[7],

pages

360-367).

Here wementionthe followingone caserelatedto this paper:

ISOTROPIC PETROV

TYPE

0SOLUTION.

T

(p

+

p)u(R) u

+

pg, f

Au

(R) u

+

Bg,

p

+

p 2AB

>

O, p 3B >0, e l

Relatingthiscase tothe equations

(3.2)

and

(3.3)

weget

1 2AB,

x=2AB-3B

2+-r

andA=0

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