Real hypersurfaces of type A in quarternionic projective space

(1)

VOL. 20 NO. (1997) 115-122

115

REAL HYPERSURFACES

OF

TYPE A

IN

QUARTERNIONIC

PROJECTIVE SPACE

U-HANGKIandYOUNG JIN SUH

Department

ofMathematics

Kyungpook University

Taegu

702-701

Republicof

Korea

JUANDEDIOS

PREZ

Departamento

de

Geometra

y

Topologa

Facultad de Ciencias

UniversidaddeGranada 18071-Granada

Spain

(Received October 6, 1994 and in revised form May 30, 1995)

ABSTRACT.

In

this paper, undercertainconditionsonthe

orthogonal

distribution

:D,

wegive acharacterization ofrealhypersurfacesoftype

A

inquaternionic projective space

Qpm.

KEY WORDS AND PHRASES.

Quaternionicprojective space, Realhypersurfacesoftype

A,

Orthogonal

distribution

1991

AMS SUBJECT CLASSIFICATION CODES.

53C15,

53C40.

1. Introduction.

Throughout

this paper

M

will denoteaconnectedrealhypersurfaceofthe quaternionic pro-jective space

QP",rn>3,

endowedwith the metric gofconstantquaternionicsectional curvature 4.

Let

N

beaunitlocalnormalvectorfieldon

M

and

Uz

-J,N,

1,2,3, where

is alocal basisof the quaternionicstructureof

Qpr,-,,

[2].

Now

let usdefine adistribution 29by

29(x)

{XET.M

X+/-Ui(x),i

= 1,2,3},

xEM,

of

areal hypersurface

M

in

Qpm,

which is

orthogonal

to the structure vector fields

{U1,

U2,

U3 }

and invariantwith respect tothe structuretensors

{

1, 2,

3

},and

by

D

+/-

Span{

Vl,

U2,

U3

}

its

orthogonal

complementin

TM.

Thereexistmany studiesabout realhypersurfaces ofquaternionicprojectivespace

QP"

(See

[1],[3],[4],[5],[6]).

Among

them Martinezand thethirdauthor

[4]

haveclassifiedreal hypersurfaces

of

QP"

with constant principal curvatures and the distribution 29 is invariant by the shape operator

A. It

wasshown that these realhypersurfacesof

QP’

couldbe dividedintothreetypes

whicharesaid to beof type

A1

,A2,

and

B.

Withoutthe additional assumptionofconstantprincipalcurvatures,as afurtherimprovement

ofthisresult Berndt

[1]

showedrecently thatall realhypersurfaces of

QP’

also could bedivided

intothe above threetypeswhen two distributions 29and 29+/- satisfy

g(AD,

29+/-)

O.

Moreover,

it is knownthat the formula

g(A29, D

+/-)

0 isequivalent to the fact that the distribution

D

is

invariantbytheshape operator

A

of

M.

(2)

hyper-sphereoratubeover atotally geodesic hyperplane

Qp,.-I

andoftype

A2

denotesatubeover a totally geodesic quaternionic projective space

QP’

(1

<k<m

2)

respectively.

Moreover,

real

hypersurfaceoftype

B

denotesatubeoveracomplexprojective space

CP".

Now,

let usconsider thefollowingconditions that theshapeoperator

A

of

M

in

QP"

may

satisfy

(VxA)Y

3

-=,{/,(r),x

+

g(

,z, r

)v,

},

(.)

g((A,-

,A)X,Y)

0,

for any 1,2,3,andanytangent vector fields

X

and

Y

of

M.

Pak

[5]

investigated the aboveconditions andshowedthat they areequivalent toeach other.

Moreoverhe used the condition

(1.1)

tofind alower bound of

IIVAII

forreal hypersurfaces in

QP".

In fact,

it wasshownthat

[[VA[[2>_24(m-

1)

for such

hypersurfaces

and theequalityholds if andonlyif the condition

(1.1)

holds.

In

thiscase it wasalsoknown that

M

islocally congruent

toarealhypersurfaceoftype

Am

or

As,

whichissaid to beof type

A.

Ifwe restrict the properties

(1.1)

and

(1.2)

to the

orthogonal

distributionT), thenfor any

vectorfields

X

and

Y

inZ)theshape

operator A

of

M

satisfiesthefollowingconditions

(VxA)Y

-Ei=lg(,X,Y)U

and

g((A,

iA)X,Y)

0

(1.4)

forany 1,2,3. Thus the aboveconditions

(1.3)

and

(1.4)

areweakerthantheconditions

(1.1)

and

(1.2)

respectively. Thusit is natural that real

hypersurfaces

of type

A

shouldsatisfy

(1.3)

and

(1.4).

From

this point ofview wegiveacharacterization ofrealhypersurfaces of type

A

in

QP’

asthefollowing

THEOREM.

Let U

be areal

hypersurface

in

Qpr,,

m>_3, satisfying

(1.3)

and

(1.4)

for all

X,

Y

in

:D

and any 1,2,3. Then

M

is

congruent

to anopensubset ofatube ofradius

roverthe canonically

(totally

geodesic)

embedded quaternionic projectivespace

QP’,

forsome

k{0,1, .,m-1},where0<r<

2. Preliminaries.

Let

X

beatangentfield to

M. We

write

J,X

iX

+

f,(X)N,

1,2, 3, where

,X

is the

tangent component of

JiX

and

f,(X)

g(X,U,),i

1,2,3. As

J

-id, 1,2,3, whereid

denotesthe identityendomorphismon

TQP",

weget

2,X=-X+f,(X)U,,

f,(,X)=0,

,U,=0,

i=1,2,3

(2.1)

forany

X tangent

to

M. As JJj

-JjJ,

J,,

where

(i,j,k)is

acyclicpermutationof

(1,2,3)

weobtain

iX

Cj,X

h(X)U

-kX +

fj(X)U,

(2.2)

and

f,(x)

f(,x)=-h(,x)

(2.3)

forany vector field

X

tangentto

M,

where (i,j,k)isacyclicpermutationof

(1,2,3).

It

is also

(3)

(,x,

-)

+

a(x,

,Y)

0,

(,x,

,y)

a(x,

Y)

f,(x)f,(Y)

(2.4)

and

,u:

=-+u,

u

(2.5)

(,j,

k)

being a cyclic permutation of

(1,2, 3).

Fromthe expression of the curvature tensor of

Qpm,

m>2, wehave that the equations ofGaussand Codazziarerespectively given by

R(X,Y)Z

=g(Y,

Z)X

g(X,Z)Y

+ ,3=,

{g(,y, Z),X

g(,X,Z),Y

+

2g(X,

,Y),Z}

+

g(AY,

Z)AX

g(AX, Z)AY,

(2.6)

and

(VxA)Y

-(VyA)X

,3=a{f,(X),Y

f,(Y),X

+

2g(X,,Y)U,}

(2.7)

forany

X,

Y,

Z

tangent to

M,

where

R

denotes thecurvature tensorof

M,

See

[4].

From

theexpressionsofthecovariantderivativesof

J,,

1,2, 3,it iseasy tosee that

VxU,

--p,(X)Uk

+

pk(X)U

+

iAX

(2.s)

and

(Vx,)Y

-p.(X)kY

+

p:(X).iY

+

f,(Y)AX

g(AX,

Y)U,

forany

X,

Y

tangent to

M,

(i,

j,

k)

beingacyclic permutation of

(1,

2,

3)

andp,,

local 1-formsdefinedon

M.

(2.9)

1,2, 3,

3. Proofofthe Theorem.

Let

M

be areal hypersurface inaquaternionicprojective space

QP"’,

andlet 7)be a dis-tribution defined by

7)(x)

{XET,

M

X..kU,(x),i

1,2,3}.

Now

we prove the theoremin

the introduction.

In

order to prove this Theoremwe should verify that

g(AT), D

"t)

0 from the conditions

(1.3)

and

(1.4).

Thenbyusing atheoremofBerndt

[1]

we can prove that areal hypersurface

M

satisfying

(1.3)

and

(1.4)

is locally congruent to one oftype

Aa,

or

A2

inthe

Theorem.

Namely we can obtain another new characterization ofreal

hypersurfaces

of type

A

in a quaternionic projective space

Qpm. For

this purpose weneed a lemma obtained from the re-stricted condition

(1.4)

asthefollowing

LEMMA

3.1.

Let

M

bearealhypersurfaceof

QP"’.

Ifit satisfies the condition

(1.4)

for all

X,

Y

in7)andany 1,2,3, then wehave

9((VxA)Y,Z)=g(AX, Y)g(Z,V),

i=1,2,3,

where 5 denotes the cyclicsum with respect to

X,

Y

and

Z

in7) and

V,

standsforthevector field definedby

,AU,.

PROOF.

Differentiatingthe condition

(1.4)

covariantly, foranyvectorfields

X,

Y

and

Z

in

D

we

get

(4)

Now

let usconsiderthefollowingfor acasewhere 1

g((VxA)Y,

,

Z)

+

g((VxA)Z,

1

Y)

-g((Vxl)];

AZ)

g(lVxY,

AZ)

+

g((VxI)AY,

Z)

g(AVxr,

1Z)+

E,lgi(Y)g(,AX,

Z),

where

g((Al

-IA)Y, U,)

isdenotedby

O,(Y)

andwehaveused the fact that

g((Al

aA)Y, VxZ)

E,t,(Y)g(V,,

V-Z)

-EiO,(Y)g(,AX,

Z).

Then by takingaccountof

(2.8)

and

(2.9)

and usingthe condition

(1.4)

again, wehave

g((7xA)Y,

IZ)+g((7xA)Z,

1Y)=

fa(AZ)g(AX,

Y)

+

(AY)g(AX,

Z)

+

EiO,(Z)g(,AX,

Y)+

E,O,(Y)g(,AX,

Z).

(3.2)

In

this equationweshall replace

X,

Y

and

Z

in 29cyclically andweshallthenadd the second

equation to

(3.2),

from which we subtract the third one. Consequently, by means of Codazzi equation

(2.7)

we get

g((VxA)r,

1Z)

=fl(AZ)g(AX,

Y)

+

EiO,(X)g(A,Y,

Z)

+

E,lg,(Y)g(A,X,Z).

From

this,replacing

Z

by

1

Z,

wehave

g((VxA)Y,

Z)

=g(V, Z)g(AX,

Y)

E,t?,(X)g(A,Y,

Etg,(Y)g(A,X,

1Z).

(3.3)

where

V1

denotes

1AU1

and the second term of theright side aregiven bythefollowing

EiO,(X)g(AiY,

1 Z)

g(X,

1AU1

)g(AY,

Z)

+

{g(AlX,

U2

+

g(AX,

Vs)}g(AY,

CsZ)

{g(AIX, Us)

g(AX,

U2)}g(AY, 2Z),

from this,the third termcanbegiven by exchanging

X

and

Y.

Thus substitutingthis into

(3.3),

wehave

g((VxA)Y,

Z)

hg(Vl,

Z)g(AX,

Y)

+

a(X,

Y,

Z)

+

a(Y, X, Z),

(3.4)

where denotesthe cyclicsumwithrespectto

X,

Y

and

Z

in 29and

a(X, Y,

Z)

denotes

-{g(AIX,

U2

+

g(AX,

Us

)}g(AY, bsZ)

+

{g(ACaX,

Us

g(AX,

U:)}g(AY, 2Z).

Then byvirtueof the assumption

a(X, Y, Z)

isskew-symmetricwithrespect to

Y

and

Z

in 29.

Now

firstly let ustake cyclic sumof the bothsides of

(3.4)

one more time.

a(X,Y,

Z)

to the right and the equation ofCodazzi

(2.7)

to the left of the

obtainedequation respectively,we have the above result for 1.

For

acasewhere 2 or 3

by using thesamemethodwecanalsoprovetheabove result.

PROOF

OF

THE THEOREM.

From

the assumption

(1.3)

we know that the shape

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REAL HYPERSURFACES IN QUATERNIONIC PROJECTIVE SPACE

(v,z)(Ax, r)

+

(V,Y)(AZ, X)

+

(Y,

X)(AZ, Y)

O.

(3.5)

Thusinorderto prove

g(AD,

:D

+/-0,wesupposethat thereisapointp at which

9(AD,

2)

+/-)pO.

Then there exists a neighborhoodL/

{pEM

g(AD,

D+/-)p#O}

on which there exist such a distributionD. Nowlet usdenote

AU,

by

AU,

W,

+

Eja,jUj,

(3.6)

where W,,i 1,2,3 denote certain vectors in 9. Since on this neighborhood L/ we have

g(AD,

:D

+/-):ri0,

at least one of the vectors

W,,

1,2,3 should not be vanishing. Thusfor a conveniencesake let us assumethat

W1

isa non zero vectoron this neighborhood/g. Then it

follows that

V1

A

U1

lWl

--

E

jo

U.1,

W1E

),

sothat,

(3.5)

gives thefollowingforany

X,

Y

and

Z

in9:)

g(IW1,Z)g(AX,

Y)

+

g(IW,Y)g(AZ,

X)

+

g(,WI,X)g(AZ,

Y)

O.

From

this,putting

Z

1W1, thenfor any

X,

Y

inT)

IIwII2

g(AX,

Y)

+

g(a W, Y)g(AI

W1,X)

+

g(I

W1,X)g(AI

Wa

Y)

O,

(3.7)

sothat, putting

Y

1Wa

gives

2[IWa

I[=

g(AX,

W

+

g( W1,X)g(AI W1,

1

WI

O.

(3.8)

From

this,putting

X

1

W,

by virtueof

IIW

II#0

wehave

g(Al

W,

1 W1

0.

From

thistogetherwith

(3.8)

wehave

g(AX,

1

W1

0.

for any

X

inZ). Thusit canbewritten

AI

W

).1..

From

thistogether with

(3.7)

itfollowsthat for any

X,

Y

in Z)

g(AX,

Y)

O,

wherewealso have used the fact

IIW

II#0

onaneighborhood

H.

Unlessotherwisestated let us continue ourdiscussiononthis openset//. Accordingly, by

(3.6)

weknowfor anyXe/)

AX

E,g(AX,

U)U,

Eig(X,

AUi)U

(3.9)

(6)

(VxA)Y

3

-,=lg(,X,Y)U,

(3.10)

"1

(X, Y)U,

+

A=(X, Y)U:

+

Aa(X,

r)ua.

for any

X,

Y

in7). Sincewe haveput

AU1

Wj

+

E.a.U,

fromwhich itfollows

(VxA)U

=VxW

+

E.oi.{-pk(X)U,

+

p,(X)Uk

+

A{-p2(X)Ua

+

pa(X)U2

+

Then forany

X,

Y

inT)the function

(X, Y)

is givenby

l(X, Y)

=g((VxA)U,,

Y)

=g(VxW,Y)

+

.ai.g(jAX,

Y)

+

p2(X)g(AUa,Y)

(3.11)

pa(X)g(AU,

Y)

g(A

AX, Y).

Whenweput

X

W

and

Y

W

in

(3.10),

weget

(W,

W)

-IIW

=-

(a-)

On

the other

hand,

by theequationofCodazzi

(2.7)

and using

(3.6)

and

(3.9)

wehave

(vv A)W

(V

A)U

W

=Vu,

(AW

AVu,

W

Vw,

(AU

+

AVw,

f

=,Vl(g(Wi,

W1

))Vi

+

ig(W,,

Wl )Vv,

Vi

2,{-(w)v

+

p,(w)u

+

AW

}

+

A{-p(W)Us

+

pa(W

)V

+

1AW1 }.

From

this, substituting

(2.8)

and taking theinnerproductwith

W

and using

(3.6),

wehave

g(v,

w,

w)

=llw II(llw111

)

(AVv, W,

W1)

2ag(AW,

W)

p2(Wl

)g(AV3,

(1Wl

--

p3(Wl

)g(AU,

1 W1

+

g(A

AWl,

W

).

(3.13)

On

theother

hand,

it canbe easilyverifiedthat

g(AVu,

Wl

1Wl

--g(Vv, Wl

A

W,

=(w,,

w)(vv

w,

v,)

Eg(W,,

W

)g(W,

,AU)

--0

wherewehave used

(3.9)

and

(2.8)

tothe secondand the thirdequality respectively.

Moreover,

the factsthat

AW

E,g(W,,

W)UT)

+/-and

W

7) imply

jojg(jAWl,

1

W1

0.

(3.14)

(7)

g(Vw,

w,

3)

=llW II(llW

p(W

)g(AU,

W

(3.15)

+

p3(W1

)g(AU2,

1 W1

+

g(AIAH/’,

1 W1 ).

Now putting

X

W1

and

Y

1W1

in

(3.11),

from whichsubstituting

(3.15)

and using

(3.14),

wehave

/1

(Wl,

1

W,

IlWl II(llW,

1).

From

this and

(3.X2)

weknow

IlWall

0, whichmakes contradiction onb/. Using the same method for the caseswhere

W2

or

Ws

arenonvanishing,wecan also prove

We

0 or

Ws

0

respectively. This makes a contradiction.

From

this we know that there does not exist such a

neighborhood/a on

M.

Thuswe can conclude

g(A79, D

-L)

0. Then from

[1]

M

iscongruent to anopen part of either a tube of radius r, 0

<

r

<

overthe canonically

(totally

geodesic) embeddedquaternionic projective space

QP’,

k{0,1,

rn-

1}

oratubeof radius r, 0

<

r

<

,

overthe canonically

(totally

geodesic) embedded complexprojective space

CP

m.

Let

usconsider the second kind oftubes. The principal curvatures on:DJ- and Z)of such a tubearegiven byal

2cot2r,

2

as

-2tan2r,

A

cotr _{and I} -rant, withmultiplicities 1,,

2,2(m

1)

and

2(m

1)

respectively

([1],[4]).

Moreover,

it isalsoknown that

A,

+

2

A,X

O,X,

i=1,2,3

2A

,

foraprincipalvector

X

in

:D

withprincipalcurvature

A.

Whenweconsiderforthecaseswhere 2

as

-2tan2r, wehave

(A,- iA)X

-(cotr

+

tanr)iX,

2,3

for any

X

in

D

withprincipalcurvature

corr.

Then from

(1.4)

wehave -rant cotr 0. This

implies that

cot2r

-1,whichisimpossible. Thus thesecondkindoftubescannot satisfy

(1.4).

Thiscompletes the proof oftheTheorem. O

ACKNOWLEDGEMENT.

The first and second authors were supported by the grants from

TGRC-KOSEF

and

BSRI

program, Ministry of Education,

Korea,

1995, BSRI-95- 1404. This work was donewhile thesecond authorwasa visiting professor ofthe Universityof

Granada,

SPAIN.

The present authors would like to express their sinceregratitude to the referee who made

someimprovementsin theoriginalmanuscript.

1.

BERNDT,

J.

(1991),

9-26.

2.

ISHIHARA,

S.

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