On normally flat Einstein submanifolds

VOL. 20 NO. 3 (1997) 497-502

ON NORMALLY FLAT EINSTEIN SUBMANIFOLDS

LEOPOLDVERSTRAELEN

K.U.Leuven

Department

ofMathematics

Celestijnenlaan 200B 3001Leuven,Belgium

and

GEORGES ZAFINDRATAFA

Universit6deValenciennes Institut desSciences etTechniques

B.P.311, F-59304 Valencienes,Cedex, France

(ReceivedDecember 21, 1990andin revised formApril 2, 1993)

ABSTRACT. Thepurposeofthispaperis tostudy thesecondfundamental form ofsome submanifolds

M

in Euclidean spacesE" which have

flat

normalconnection. As such, Theorem gives precise expressions for the (essentially

2)

Weingartenmapsof all4-dimensional

Emstem

submanifoldsin

E

6,

which arespecializedinCorollary2 tothe

Rcciflat

submanifolds. Themainpart ofthispaperdealswith

fiat

submanifolds. In 1919, E. Caftan proved that every flat submanifold"of dimension

<

3 in a Euclidean space istotally cylindrical.

Moreover,

heassertedwithout proofthe existenceof flat non-totally cylindrical submanifolds of dimension

>

3 in Euclidean spaces We will comment on this assertion, andin thisrespectwillprove, inTheorem3,thatevery flatsubmanifold

M"

withflatnormal connectioninlg istotally cylindrical(forall possible dimensionsnandrn).

KEY WORDS AND PHRASES. Submanifolds, normal connection,Ricciflatsubmanifolds. 1991AMSSUBJECT CLASSIFICATION CODES. 53c25.

1. INTRODUCTION.

This paper deals first of all with the second fundamental form ofan Einstein submanifold of codimension 2.

ARiemannianmanifold is Einstein if its Ricci tensor field isproportional(witha constant coefficient of proportionality)to the Riemannian metric. We recall that every space ofconstantsectionalcurvature

is Einstein

The converse statement is true also in 2 and3 dimensions, as shownby J.A Schouten and D.J Struik in 1921

FACT

A (see

lSl

or

[51

or[1]). IfaRiemannianmanifold

M

of dimensionn

(n _< 3)

isEinstein, then it is aspaceofconstant curvature.

T.Y.Thomasin 1936andA.Fialkowin 1938classifiedtheEinsteinhypersurfacesofthereal space forms Inparticular,wehave

FACTB (see [9] or [6]or [10] and [1]). Let M beahypersuffaceimmersed inE

n+l,

where

n

_>

3. If

M

isEinstein,then:

(B. 1)theRiemannianscalar curvature, say8,of Misconstantand non-negative,

(B.2)

if8 0, then

M

islocallyEuclidean;

(B.3)

if8

>

0, then every point of

M

isumbilicaland

M

islocallyahypersphere S’.

FACT C (see [9]or[1]). Let

M

bea Riemannian4-manifold. Then

M

is Einsteinifand only if, foreverym E

M,

forany2-plane

P

atn, thesectional curvatureof

P

isequaltothe sectionalcurvature of the2-planeP+/- perpendicularto

P

atn.

Themethod of theproofof Theorem inspiresus to establishinTheorem3 a relationbetween flatness and cylindricity. The importance ofthis relation willbe justifiedinFact D.

2. STATEMENTSOF THE MAIN RESULTS.

THEOREM1. LetMbe a 4-manifoldisometricallyimmersed with flat normal connection in

E

.

Then

M

is Einstein ifandonlyifforeach pointmEM:

(1.1) either

M

iscylindricalatm;

(1.2) orMisumbilical(non-geodesic)withrespectto anormaldirection

N1

atmand cylindricalin anothernormal direction

N2

perpendicularto

N1

atm;

(1.3) or with respect to a suitable onhonormal tangent frame of

M

at m and an orthonormal normal frame

{N1,

N2}

at

,

theWeingarten operators

AN1,

AN,

admitrespectively oneamongthe followingmatricialrepresentations:

ANt=

0_{0 0}b 0₀ 0₀

AN,=

0_{0 0}0 0_c 0₀

0 0 0 0 0 0 0 d

(1.3.1)

where ab cd;

a 0 0 0

1

0 a 0 0

AN

_{0 0 --a 0}

0 0 0 --a

(1.3.2)

whereaisanon-zero realnumber,and

N2

iscylindrical;

(1.3.3)

where 4-1, a,b,p,q arereal numbers such thatab 0 and pq

e(a

);

b2

0

-

0 0 0 p 0 0

(1.3.4)

Av,

_{0 0 0}

Av,=

_{0 0 q 0}

a ₀

00’u

0 0 0

where a,b, c, d,p,q,uarereal numbers suchthata

-

0, and

pq=d-

,

pu=c-

,

qu b-

,

and

(b-

cd bd

(d-

>

o.

With respect to case 1.3.1of Theorem 1,wegiveinparticularthefollowing

EXAMPLE AND REMARK 1. Let

Ml(c)

and

M2(c)

be two surfaces ofconstant Gauss curvature cintheEuclidean3-spacelg

3.

Then

(1)

theRiemannianproduct

M

4 _M]

(c)

_x

M2(c)

canonically isometricallyimmersed inlg is an Einstein2-dimensionalsubmanifold withflat normal connection. Itis not aspaceofconstant curvature andmoreover it is not Riccifiat, unlessc 0.

(2) Inparticular, forc

<

0, forinstance

M](c)

and

M2(c)

both beingapseudo-spherein

E

of the same pseudo-radius c, the Riemannian productmanifold

M

is an Einstein submanifold with flat normalconnection in

E

6_{whichhasstrictly negative scalarcurvature.}

COROLLARY 2. Let

M

bea 4-dimensional manifoldisometricallyimmersed with flat normal connection inlg

.

ThenMis Ricciflatifandonlyiffor eachm E

M;

(2.1) either

M

isflat(hencecylindrical)atm;

(2.2)

or with respect to a suitable orthonormal tangent frameat rn and anorthonormal normal frame

{N

a,

N2)

at m, the Weingarten operators

AN

,AN

admit respectively one ofthe following matricialrepresentations:

0

-

0 0 0 p 0 0

(2.21)

AN,=

_{0 0}

-

0

AN=

0 0 q 0

0 0 0 a 0 0 0 0

wherepq a

>

0.

0 g 0 0 0 p 0 0

(2.2.2)

A..,v.

_{0 0 0}

A..,

0 0 q 0

0 0 0

.

0 0 0 u

wherea

and++a

o,(a-

).(-

).

(b-

)>o.

TIIEOREM 3. Let M be a n-dimensional manifold isometrically immersed with flat normal

cbnnectionin

E

+N.

Then

M"

isflatif andonly ifit iscylindrical. 3. DEFINITIONS [3].

We consider a manifold

M

isometrically immersed with codimensionN in theEuclidean space

]n+N.

3.1. Let beanormal

vector.field

onM.

Weshall say that

M

isquasi-umbilicalinthe direction ifthe Weingartentensor

A

of admits an eigenvalue

A

withmultiplidtyn Iorn.

Inparticular:

(i) if

(ii) if

3.2.

M

is(totally) cylindrical[resp.quasi-umbilical] if,locallyaround each point, thereexists an orthonormalnormal framefieldcomposedwithcylindrical[resp.quasi-umbilical]directions.

Nowweproveourresults. 4.1 PROOF OF

THE

THEOREM1.

LetMbe a 4-manifoldisometricallyimmersed in the Euclidean6-space

Supposethat

M

is Einstein. Thenby Fact

C,

forany m E

M

and any 2-plane Pin

T,M,

its sectionalcurvature isthesameasthesectional curvatureofitsorthogonal 2-plane

px

in

T,,M.

To exploit thisstatement, we supposemoreoverthat the normal eormeetionof

M

in E is fiat.

Then, at each point m

M,

there exists anorthonormal tangent frame

{el(m),

,e4(m)}

which diagonalizes simultaneously all

W.eingarten

tensors of

M

(at m). We denoteby

cq(m)

the sectional curvatureof the2-plane

{e,(m),ej(m)}

for1

<

<

j

<

4. Then

M

isEinsteinian if andonlyiffor each

mM.

cgt(m)

C34(m

ca(,)

c4(r-)

(*)

c4(,)

().

Nowwefixthe pointminM. Either

M

is geodesic at m: then theproblemissolved. OrMis non-geodesicat m; we can assumethat

a,(el(m),ea(m))

0 where cr, isthe second fundamemal

o.,(e(,,,)x(,))

and denote

N2

theunit normalvectorperpendicularto form atm. We canput

N1

o,.(e(m),e(m))ll

N.

By

our choice of the tangent frame

{e(m),.-. ,e(m)},

the Weingarten tensors

Av,Av:

relative toN1,

N

respectively can berepresented by thematrices:

Au

0_{0 .0}Ag.

Az

0 0₀

A=

0₀

=

_{0 #3}0 ₀0

Hencetheprevioussystem(*)isequivalentto thefollowingone:

i2

b

(1)

c

(2)

AA4

d

(3)

(**)

A3A

+

3 b

(4)

A=A

+

## c

(5)

A=A3

+

=

d

(6),

where b c2 c34,c c13 c24,d c14 c23.

Toresolve

ts

systemof6equations th7

unos,

let us firstcompute

A1

Using the equations

(1), (2),(3)dthe equityb

+

c

+

d

s

(wheresistheconsttscgcuatureofM)wefindthat

A1

is alutionoftheequation:

z 4

< H,

N

>

z

+

s

0

(***)

where x is

uo

md Histhemcuature vectorat m. Such mequation adts a solution A] asince:

4

<

H,N]

>

s

(A1

+

A2

+ Az +

A4)

A(A=

+ A +

A)

0.

Todetenethe

uos

A2, A3, A4,p2,

,,

we discuss onthe index of

nomulliW

r(m)

of

M

at m, i.e., the

r

of the

emm

amre

operator at m. Bause ofthe system (*),

()

e

{0,

2, 4,

}.

CASEl:

=(m)

O. Then

M

isflat(hencecci

flat)

atm. Bythesystem(**),

M

is cyfinddcatm.

CASE2:

(m)

2. Thenweobtn the situation

(l.3.1).

CASE3:

(m)

4. Itis tochk that

ts

isimpossible.

CASE4:

=(m)

6. Fromasimplediscussion on the

r

of

AN,

wededuceeither

(1.2)

or

(1.3.2)or(1.3.3),or

(1.3.4).

Tsproves the Theorem

4.2.

.

Inacrdmce th ch of the possibilities

om

Theorem md

Coroll

2,wecmngmctloc pettion of submfolds ofcodimeion 2 in

Eo

th flat

no

come,ion wch e, at a pil poim,Eingeor pmiculcciflat.

5. ON

T

SOLDS.

5.1. A flat mold is in picul Einein.

In

1919 [2], Erie Cm studi the second

ndemfo of flat submolds of aEucfidemspace.

FACT D. ([2]).

.1)

Eve

n-mensionflatmbmfold;dofE"+N_{thn 3 is}

linddc.

Moreover: .2) E.Cmtedthoutproof,that the

ason

.

l)flsifn 4.

With

respe=

to.2),weconsiderthe case ofdimension n 4.

sume

h:

E

x

E4

EN

is aflat bilin

c

pmd consider themensionof the

vor

space

[Imh]

generated by

e

age

of h. We may suppose thout loss of generity that

N

dim[Imh].

Sin themensionofthespaceof

sec

bfoson

E

is

equ

to10, wecm

re

oselvesto0 N 10. Using

tecques

for theproofofFa

.

l),it issyto

demonstrate

tt,

ifN6

{7,

8,

9,10},

wecmreduceNsothat N6

{0,1,

2, 3, 4,5,

6}.

Inthe se

parwhereE. C

prov

Fa

.

1),he showed

so

thatfor the ceN6

{

0,1, 2, 3,

4}

the flatness implies

e

cyfinddci. Consequemly, the

oy uo

ces e: "N 5"md"N 6". In1986[7],

exple of a4-mbmold in EI _{wchis, at apicul poim, flat thoutbeing}

finddc

is

constm. However,

all justifition ofFact .2)is=illlacing for themoment; inother wordsthe methodofresolutionof the so-clGaussequation ofaflat submfold in a Eucfidemspaceisstill

uo

inmensionndincodimensionNthN n

+

1, evenfor thecaseof dimensionn 4.

Onefirst resolution for sucha problem is

ven

in Theorem 3 for the piculce of flat

no

come=ion.

5.2. PROOF OFEOM3. For

ts

puose,weapplythefollong Fact EdLena(*)

wchwe atmdprovebelow:

FACT E

(s

[7]and[2]). Letvbea

veor

space, letwbe mother

veor

space,dowed tha

scproduct <-,. >.

Suppose

:v

xvw is a bifin

setdc

map, flat th respect to

<

-,-

>

(i.e.,

< (x,

y),

(z, w)

>

< (x,

w), (y,

z) >

for y x,y, z,w in v.

sume

moreover that the

onhogoprojtion of on asubspace Wofwis

lindfi.

LEMMA (*). Let a:E xE EN _{be a flat bilinear symmetric}

map satisfying the following property (F): "There exists an’orthonormal frame

B

{el,...,

e,}

in

E"

which diagonalizes simultaneouslyallprojections

<

a,

>

ofainanydirection EE

v’’.

Thenaiscylindrical.

PROOF. Weshallprovethislemma byinduction on

N,

andsupposecis notgeodesic. Thelemma is treeforN 1.

Considerthe caseN 2. Let

{1,

:}

beanorthonormal frameinIg

.

_{The property(F)implies}

that eachcomponent

<

a,

>

canbe representedinthe frameBbythe matrix

0

<

,

>

,

0 ".

Thesectional curvature ofeach2-planegeneratedby

{e,

e}

isgiven by

We may supposethat

a(el,

el)

:fi

0 and

f2

iscollinearto

a(el,

el). By

this manner:

A

0 and

A2

0. Sinceaisflat,the arebothnull. Thisimplies:

A,

0for2

_<

_<

n.

Henceais flatand

<

a,

f2

>

iscylindrical. ByFactE,

<

a,

f

>

ifflattoo. Since

<

a,

f

>

is a

(flat)bilinear symmetric form, it iswell-known that it is cylindrical. Hencethe lemma isproved for N=2.

Now suppose Lemma

(*)

is treefora certaininteger k andanydimensionNwithN

<_

k. Letus

provethatitthenis also trueforN k

+

1.

By

ourhypothesis,a:Ig x]g ]g+1 _{ifflat andenjoys}

the property

(F).

Whenwe reasonas forthecaseN 2,weeasilyfindthat

<

a,

+1

>

iscylindrical. We applyFact Eagain anddeduce that theprojectionaonthehypersufface]g ofEk+l_{perpendicularto}

+1

isflattoo,a:

E

x

E

lg

.

Ourhypothesis ofinductionobviouslyassertsthataiscylindricaltoo. Hencea:

iscylindrical. ThiscompletestheproofofLemma (*). I-I 6. OPENPROBLEMS.

Forthe moment,the following questions relatedtothispaperremain still without answer.

PROBLEM1. Howtoclassifyall Einstein4-manifolds, andinparticularall Ricciflat 4-manifolds?

(see [1

]).

PROBLEM2. Resolve theGaussequation ofaflat submanifold

M

of codimension5 or 6 inthe Euclidean space i.e.,findallbilinearsymmetricmapa"

4

4

_,

Ev

(for N 5or6)satisfying the equality

<

a(z,

y),

(z, w) >

<

a(z,

z),

a(, w) >

0for any z,

,

z,win

4

(consideronly the casewhen the kernel Ker of istrivial!).

ACKNOWLEDGEMENT. The second author would like to thank Professor Abdus Salam, the International Atomic

Energy Agency

and theUNESCO for hospitalityatthe]International Centrefor TheoreticalPhysics, Trieste, wheremostofthe workon thisarticlewasdone. Thepaperwas finished when the second authorwas aDoctoral ResearchFellowattheKatholiekeUniversiteitLeuven.

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EinsteinManifolds, Ergebnisseder Mathematik and ihrerGrenzgebiete,

Springer-Verlag,Berlin.

4E,

Surles

vari6tds

de courbureconstated’unespaceeuclidien ou noneuclidien, Bull. 2. CARTAN,

,(

Soc.Math. 1919), 25-160.

3. CHEN,

B.Y.,

Geometry ofSubmanifolds, M. Dekkar, New York, 1973

4.

FIALKOW, A.,

Hypersurfacesofaspace ofconstantcurvature,Ann.

of

Math. 39

(1938),

762-785. 5.

KOBAYASHI,

S.

& NOMIZU, K.,

Foundations

of

Differential

Geometry,

Wiley,Interseience 1,

1963.

6. KOBAYASHI, S.

&

NOM/ZU,K., Foundations

of

Differential

Geometry, Wiley,Interscience2, 1969.

7.

MORVAN,

J.M.

& ZAFI

RATAFA,

G.K., Conformally fiat submanifolds, Ann. Fac. Sci. Toulouse,VIH.3(1986-87),331-347.

8. SCHOUTEN, J.A.

& STRUIK, D.J.,

Onsomeproperties ofgeneralmanifolds relatingtoEinstein’s theory of gravitation,Amer. J.Math.43_(1921),213-216.

9.

SINGER,

I.M.

& THORPE, J.A.,

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Papers

in

Honour

ofK.

Kodaira,PrincetonUniversity

Press,

Princeton

(1969),

355-36-5.