Unique isometric immersion into hyperbolic space

Internat. J. Math. & Math. Sci. VOL. 16 NO. 4 (1993) 725-732

725

UNIQUE

ISOMETRIC

IMMERSION

INTO

HYPERBOLIC SPACE

M. BELTAGY

Mathematics

Department

Facultyof Science

Tanta

University

Egypt

(Received

February6, 1992 andinrevisedform

August

24,

1992)

ABSTRACT.

A

result concerningequivalenceofisometric immersions withrelated

Gauss

maps intohyperbolicspacehas been established.

KEY

WORDS AND

PHRASES.

Isometric immersion, rigidity, second fundamental tensor,

Lorentz

manifold,Minkowskispace.

1991 AMS

SUBJECT

CLASSIFICATION CODES.

53C40, 53C42.

1. INTRODUCTION.

Equivalencebetween isometric immersions ofa Riemannian manifold into another one has been aninteresting fruitful areaof research for alongtime

([1], [4], [6], [7], [12]).

The concept of equivalence between twoisometricimmersions may begivenasfollows.

Let

fl:M--N

and

f2:M--,N

be two differentisometric immersionsofaRiemannian manifold M into another Riemannian manifold N. The immersion

J’l

will be equivalent to

12

(up to an isometry rand

N)

if thereexists anisometryr:N-Nsuch that r

J’l

)’2"

Wemaywrite

J’l

I2

(rood. r).

Ifthe above mentioned concept is satisfied for each pair ofisometric immersions ofM into N,wesaythat M isuniquely isometrically immersed inN.

In

classical differentialgeometryMis said toberigid.

If for each pair ofisometric immersions

fI:M--,N

and

’2:MN"

there exists a continuous curve rs, 8e[0,hiinthe groupGofisometriesofNsuchthat r_o idandra

1 2’

wesaythat

M isuniquely continuouslyisometricallyimmersedinto N

[8].

In

[1]

thefollowingremarkable result concerningisometric immersions into(n

+

1)-Euclidean

space En

+

has beenproved.

THEOREM

1.1. Let M be a connected, orientable C-Riemannian n-manifold.

Suppose

that the

Gauss

mapsof each pair j’ and of isometricimmersions ofM intoEn

+

_{differby an} isometry ofEn

+

1.

Then M isuniquely isometricallyimmersed in En

+

if anyof the following

conditionsis satisfied.

(a)

Mis compact,

(b)

Thereexistsat leastonepointmeMwhere allsectionalcurvaturesarepositive,

(c)

Thereexists atleast onepoint rneMsuchthat_1"

(or

)

isnot minimal atrn,andMhas

As far as we are concerned, the corresponding study in hyperbolic space Hn+1 has not yet been considered. Accordingly, we devote the present work to deal with the rigidity problemof

hypersurfaces inhyperbolicspace with constant sectional curvatnre asareceivingspace. The main result will be" stated

n

the next

section.

From now on all manifolds and maps are sufficiently smooth for computationstomakesense.

2. PRELIMINARIES

AND

RESULTS.

We begin this section by giving a briefword about the model ofhyperbolic space we shall usethroughout thepaper.

In

Rn+2 with the natural basis _{O, el, e2,...,}

en+

_1’ n>_1, we consider a nondegenerate

symmetricbilinearform

n+l

b(z,y)= -Zoy

o+

y

_zkyk r, yERn

+2.

k:l

The pairVn

+

2

(Rn +

2,b

iscalled the Mmkowski space.

Let

0(1,n

+

1)denote theorthogonalgroup ofb,i.e.,

where

0(1,n

+

1) {AEGl(n+2,R):b(Az,Av) b(r,y)} {A Gl(n

+

2,R):A’SA S}

0

ln+l

The fact that A’SA=S implies that det A= +l.

A

matrix

A=(aij

in 0(1,n+l) belongs to the identity component ifandonly ifdet A and

aoo

>1. Let G denote the identity component of O(1,n

+

1).

Now,

the hypersurface in Vn+2 defined by b(z,y)=-! is the disjoint union of two connected components:

Hn+I

{z:Zo>

1} and

Kn+

{z(

:Zo

< _-1}.

0(1,a

+

1) acts transitivelyon whileGacts transitivelyonHa

+

[9].

Let zeHa

+

1,

_{i.e., b(z,y) and}

zo_>1. Thetangent space

TzH

a,+ isgiven,through the identification by parallel displacement in Va

+

2,

by the subspace of all vectors aEVa

+

2 such that b(z,a)=0. Therestrictionof to

TzH

a+ is positivedefinite. Thus the formbrestricted to thetangent space at each point ofHa

+

gives rise to acomplete Riemannian metric on//a

+

which is

obviously

invariant under G.

Moreover,

(Ha+

1,b)

is of constant sectional curvatures K 1. Hence Ha

+

will be taken to be the model in which our immersions occur.

For

UNIQUE ISOMETRIC IMMERSION INTO HYPERBOLIC SPACE 727

Thehypersurface + in

v

n+:2_{definedby b(z,y) isconnected Lorentz. We call}

.

n+

the conjugatehypersurfaceofboth Hn+ and Kn+

1.

(S

_{Fig. 1).}

Figure

Let I:M--,Hn+l be an immersion of the orientable manifold M. The Gauss map

:

M

n

+

_{ofthis immersion is definedtobethe parallel translationoftheunitnormal IV{m} of} M, asahypersurfaceofHn

+

1,

at /’(m) uptothe origin0ofVn

+

2.

Hence,the

Gauss

image ofM willlieonthe conjugatehypersurface n

+

_ofHn

+

I.

Now,

westateourresult.

THEOREM 2.1. _{Let f} and

]

be two C isometric immersionsof a connected, orientable Riemannian n-manifold M(n>2) into the (n+ 1)-hyperbolic space Hn+

1C

Vn+

2.

Suppose

that the Gauss maps and

g

differ by amatrixaeG. Then with anyoneof the conditions

(i)-(iii)

below,/’and] differbyanisometryrof

(i)

Mhasatleast onepointmeMwhereallsectionalcurvaturesaregreaterthan 1.

(ii)

Miscompact.

(iii)

Thereexists atleastonepointmeMsuch that

I (or

)

is notminimal at m,and there

isnopoint in M with A 0

(or

0),

whereAisthesecondfundamental tensor.

Assume that a where is an element of G, then we can show that the immersion

f a

.:

MHn

+

has the Gaussmap of the form +/- a

.

The signdependsonwhether a is orientation preservingor not.

Hence

+ which will beused, just forsimplicity,instead

of

=o.

In

the light of the above discussions, let us take

3

and notice that if

-

we may

change the orientation to have again

3

.

In

this way, N(m)=

(m)

for every meM up to a

parallel translation in Vn

+

2.

_{Let D be the Levi-Civita connection of Y}n+2 _{and be the} induced connection on //n+

1.

Applying Weingarten equation to I(M) and (M) as immersed

hypersurfacesofItn

+

1,

wehave

,(,X 17 f, (-AX),

(2.1a)

7

,X

7,

]X),

(2.1b)

where Aand ] arethe second fundamental tensors of

I

and ],respectively, and X isa tangent

vector of M. Ifwe write

D,f,X

N in terms ofits tangent component to Itn

+

_{and the normal}

one,wehve

Dr,

XN= Vf,X N+ Sinceb(N,f.X) 0, then

I,X b(N,l(:)) b(D

f XN,f(x})+bN,I,X) 0

=

b(Df,

X N,f(z)) 0

g =0.

(2.2b)

To be consistent with the referenceswe dependon, we_{shall replace b(X,Y) by <}X,Y> in all thefollowingcomputations.

From (2.2a)

and

(2.2b),

wehave

Fromequations

(2.1)-(2.3)

wehave

As N

,

then equations

(2.4)

give

Dr,

x

N= XT

f,X N

O7,

x

N

v],

x

N

(2.3a)

D

f

x

N f,( AX)

(2.4a)

D-/

,x

f

-/*(

]x)

(2.4b)

f,(AX)

y

,(I

X)

(2.5)

uptoaparalleltranslationin

V"

+

2.

LEMMA

2.1. Aand] have thesamenull space at each point ofM.

PROOF. It is sufficient to show that AX=O if and only if iX=0. If AX=O, then

f,(AX)

0 andso

]

,(]

x)=0. Since

]

is animmersion, then

],

is injective and hence ]

x

0. Theconverseis direct.

LEMMA

2.2. A2

]2.

PROOF.

For

arbitrary vectorsXandYtangentto M,wehave

<

]2X,

y_{> < iX,4Y> <}

],(2Y)

_>

(2.6)

<f,(AX),f,(AY)> <AX, AY> <

A2X,

y>.

Hence

the result.

LEMMA

2.3. (tr A)A (tr A)A

(2.7)

PROOF. TheRicci tensorSof M,asbeingpreserved byisometries,maybewrittenas

[9]

S /(trA)A- A2

="

/(tr A )A i2

where istheRicci tensorofHn

+

I.

Then

(trA)A-A2 (trA)A ]

2.

UsingLemma 2.2,weobtainthe requited result.

LEMMA

2.4.

(i)

(trA)2 (tr2

)2

(ii)

IftrA#0at m,then2 + Aatm.

(iii)

The choice of the sign in

(ii)

is constant in each component of the open set V {m I(tr A)(m)#0}.

PROOF. Part (i).

Thisbecomes direct fromLemma2.3 whentakingthe trace of both sides of equation

(2.7).

Part (ii).

If (tr A) (m)#O, then by part

(i)

we have that (tr

2)

(m)#0. Using equation

(2.8),

wehave [(tr A)/(tr )](m) +i. Applying equation

(2.7),

weobtainthat ] +Aatm.

UNIQUE ISOMETRIC IMMERSION INTO HYPERBOLIC SPACE 729 m. Hencethe subset

v

{ml(trA)(m) 0}is anopen subsetof M.

Nowconsiderthe followingcases"

(a)

Let A at m andassume that ineach neighborhoodV of m thereexists apoint m where

A(rni)=-l(mi).

In

this way, we obtain a sequence of points

{mi}

for which

A(mi)

-

(rni).

Taking thelimitofbothsidesasm mwehave lirn

A(mi)

lirn A

(rni).

m_f-.,m rn_f-.,m

By the continuity of both A and

,

wehave A(m)=

-(m)

which contradicts the assumption andsoA nearm.

(b)

IfA

-

atm, wefollowasimilardiscussion. 3.

A RIGIDITY RESULT.

In

this section we give a general rigidity result for hypersurfaces in Hn+ similar to

Theorem 6.4p.45-46

[9].

For

notations and detailswerefer to

[9].

Althoughthefollowingresult has beenalready given in

[13],

we write its full proofas it is too important for the subsequent

results.

THEOREM 3.1. Let Mbeaconnected n-dimensional Riemannian manifold and letfand be isometric immersions of M into H

n+IcV

n+2 with fields of unit normals and

,

,respectively. If thesecond fundamental forms h and of

!

and

(with

respect to and

),

respectively,coincideon M, then thereisanisometry rofHn

+

such thatr f.

PROOF. We follow the same procedure of

[9]

and start with the local version of the theorem.

Assume

that z_o is a point of

m.

We

have two different frames

(el,e2,-..,en,,v)

and

’

2’

’

n,

,

at Zo, wherey f(z) and (z),respectively,such that

<ei,ej >

<’’i,’j>

<i,j<_n

<,ei> <,Ti> =0, <Y,_ei> <,’i> =0, <V,>

<,

> =0.

As G acts transitively on I-In+

1,

then there exists an orthogonal matrix rEG which is an

isometry of Itn+l such that r maps the frame

(el,e2,...,,y)

at _zo upon the frame

f

,g_2,

",

,

atthesamepointz_o.

In

terms of local coordinates in acoordinate neighborhood U about to, simple calculations

using

Gauss

and Weingarten equations give the following two systems of partial differential equations

i

gijy

+

k

Ffj

ek

+

hij

j

j

+

rtj

z

t

+ hij

3/=

-taJ

zt

(7)

Themaindifferencebetween these twosystems and those of

[9]

isthat thesystemshereare

non-homogeneous.

In

spite of this difference, the existence and uniqueness principle of solutions is stillworking.

Since the two systems (I) and

(T)

above have the same initial conditions at z_o, then by

uniquenesswehave

As _ej

Oy[Ozj,

_j

0-9/0rj,

and since Y(Zo) (o), we have that y on U and the local version isnowproved.

The global versionof the above theorem can be proved exactly as that ofimmersions into Euclidean spaceEn

+

[9].

Notice that forimmersions into Euclidean space En

+

1,

the required isometry isa resultant of translation in addition to an orthogonal matrix r6-0(n+1). From the above theorem, it becomes clear that in thecaseofimmersions intohyperbolicspace Hn

+

theisometryneededto establish the proof is obtained purely from an orthogonal matrix of G acting on //n

+

by

restriction.

4. PROOF OF THEOREM2.1.

In

the light of Theorem 3.1, ifwe consider V to be the component of Von which ]

a,

then thereexists an isometry r of//n

+

such that r

7

!

on

Vl.

In

the following wegiven moresignificanceofr.

Let us consider a point m6_V and take X

6-Tram

to be an eigenvector of A(=

])

correspondingtoanon-zeroeigenvalue. Then equation

(2.5)

gives

I,(x)

],(x)

(4.1)

Cgnsequently,

if(rant A)(m) n, then all theeigenvaluesofA are non-zerosandso

I,(Tm M)=

7

,(Tm M) up to a parallel translation in Vn

+

2.

Now consider the following two different cases:

Case

(i).

(m) I(m),

7

,(TinM) I,(TmM) and (m) N(m).

In

thiscase, the neededisometry ofHn

+

turns outto betheidentitymap.

Case

(ii).

I(m)

7

(m),

]

,(Tm M)//I,(TmM) and (m)//N(m).

Clearly, (m) and l(m) should be antipodal points of

.

This situation will not occur unless ]’(M)CHn

+

and

7

(M)Cgn

+

which contradictsthe assumption that both of theimmersions

!

and

7

are in//n

+

1.

Consequently,thiscaseshouldbedisregarded. Using Theorem 3.1,wehave thefollowing.

LEMMA

4.1 If

v

is a component of

v

where ] A and rank A n at one point m6-V then I(x)

7

()onV_1.

Onthe other handif A ] _{and (rant A)(m)} n, then l,(X) ,(X) V X6.TmM.

In

this case I,(Tm M) will also be parallel to ,(Tm M) and as mentioned above I(m) should

coincide with

]

(m). The isometry ofUn

+

needed is different from

tle

identity.

In

fact it is

somesortofreflection.

LEMMA

4.2. If

v

₂is a componentofVwhere] -Aandrank A n at onepoint m6-V

2 then the matrix reG which satisfies r(m) m,

r(N)

N and r(X) X V X6-T_m M has the

propertythatr

]

(z) I()onV_2.

In

thefollowing, weshallrestrict ourselves to thecaseofLemma4.1 mentioned aboveand giveahinttoshow howtodeal with thecaseof

Lemma

4.2.

Letusdefinethe real-valuedfunction FonMasfollows F(a)-- <f(a),c >,

wherecisaconstant vector. Followingcomputations similar tothatin

([9],

p.

342)

wehave that

or

UNIQUE ISOMETRIC IMMERSION INTO HYPERBOLIC SPACE 731

where A-n and 0 is he mean curvature vector

=l

(r A)N of M as a hypersurface of

//n

+

1.

Thenewoperator iselliptic and hasCocoefficients.

Weare now inaposition

o

proveTheorem2.1.

PROOF OF

PANT (i).

Le be a poin of

vhere

all sectional curvatures are greater

than -1. Then rank A rank n and A is definite at m. Let m(V where V is the componentof

v

where

,

A. Hence .(X)= f.(X) for eachtangent vector X(T_mM. From the abovediscussions,the identitymap willbe therequiredisometry which takes (v

1)

to f(V

1).

Let W be the largest open set of M containing V such that z z. Following the same

procedures of

[1]

we conclude that forevery boundary point p(Ow, thereexistsa neighborhood

Q of_l0 such that for each q Qo

v,

A at qwhich means that Q intersectsthe component

v

only.

Hence

wemay write

w

t3Qc

Vl

t3V’_2" For pointsin

v

_1,wehave_W 0 as

X

AonV_1’ and for pointsin Q- V_1, wehaveW 0. Consequently, W ron the wholeofWoQ.

Consider thereal-valuedfunction9:woQ-,Rdefinedby

g(z) <f(z)-f(z), c>. Applyingthe operatortog, wehave

Ag= A _<f(x)-f(x), c> A _</(z), c> A _<f(z), c> =n<r/,c> --n<,c> =0 on Wt

O.

Since 0 on W, then a 0 on W_{o 0 by}the uniquecontinuation principleofsecond

order elliptic operators with 6,00

coefficients

([3],

[0], []).

Hence /’(-)

7

(-) on

woo

contradicting the maximality of

w.

Consequently,

w

will bethe whole ofM andso f(x)= for all points ofM andwehave that M andwehave that M isrigid which ends theproofof part

(i).

Consideringm(V₂where

X

-A, weapplyr,givenin

Lemma

4.2,to//n

+

andcarryout theproof ta&ing

asmis invariant underr.

PROOF OF

PART (ii).

This partof the theoremmay beconsidered as acorollaryofpart

(i)

inthe followingsense.

It has been provedin

([2],

[5])

that if M is compact and isometrically immersed in Hn+

1,

thenthereexistsapoint of

m

allof whose sectional curvaturesarepositive. Applying part

(i)

we obtainthedesiredresult.

PROOF OF

PART (iii). In

this case, as trA 0at rnandrankA#0everywhereonM,the

sameprocedurecanbefollowed butthe requiredisometry ofhrn

+

will be different fromthoseof parts

(i)

and

(ii).

ACKNOWLEDGEMENT. The authorisgratefultothe referee forhisvaluable suggestions.

REFERENCES

1.

ABE,

K.

&

ERBACHER, J.,

Isometric immersions withthesameGaussmap,Math.

Ann.

AMARAL, L.,

Hypersurfacesinnon-Euclidean space, Ph.D Thesis, Univ. California,

Berkeley

(1964).

ARONSZAJN,

N.,

A

unique continuationtheorem forsolutionsfor elliptic partial

differential equationsorinequalities of secondorder, J. Math. Pures

&

Appl.36

(1975),

235-249.

4.

BELTAGY, M.,

Immersions into manifolds withoutconjugatepoints,Ph.D. Thesis, Durham Univ.,

U.K.

(1982).

5.

BELTAGY, M., A

proofofL. Amaral’s

theorem,Ind.

J.

Pure

&

Appl.Math. 14

(6)

(1983),

703-706.

6.

BELTAGY,

M.,

A

rigiditytheoreminsphere, J. Math.

&

Comp.

Sci.

(In

Press).

7.

DOCARMO, M.P.

& WARNER,

F.W.,

Rigidityand convexity ofhypersurfaces insphere, J. Diff. Geo. 4

(1970),

133-144.

8.

GOLDSTEIN, R.A.

& RYAN,

P.J.,

Infinitesimal rigidity of submanifolds,

J.

Diff. Geo. l0

(1975),

49-60.

9.

KOBAYASHI,

S.

&

NOMIZU,

K.,

Foundations of Differential

Geometry,

Vol.

&

II,

New

York, Interscience Publishers

(1963-1969).

10.

MIZOHATA, S.,

Theoryof partial differential equations, CambridgeUniv., 1973. 11.

PROTTER,

M., A

uniquecontinuationfor ellipticequations, Trans. Amer.Math.

So<:.

95

(1960),

81-91.

12.

SACKSTEDER,

R.,

Therigidityofhypersurfaces,

J.

Math. Mech.

II (1962),

929-940.

1.

SASAKI, S., A

proofofthefundamental theoremofhypersurfacesinaspace-form,

Tensor,

N.S. 24

(1972),

363-373.