有限型部分多様体および二重調和部分多様体の分類 問題

九州大学学術情報リポジトリ

Kyushu University Institutional Repository

有限型部分多様体および二重調和部分多様体の分類 問題

石川, 晋

https://doi.org/10.11501/3063847

出版情報：Kyushu University, 1992, 博士（理学）, 論文博士 バージョン：

The Classification Problems of Finite Type

and Biharmonic Submanifolds

The Classification Problems of Finite Type and Biharmonic Submanifolds

1£W:k~ JII~$ It~

:fi Jr( if

SUSUMU ISHIKAWA

Department of Mathematics, Saga University 1034 Honjyo-machi, Saga 840, JAPAN

1992$6JJ~lli

Cont nts

Preface 3

Chapter I. In trod uctiou 5

§1.1. Ch n's Conjecture Cone ruing Biharmonic Submanifolds 5

§1.2. Chen's Conjecture Concerning Finite Type Surfaces in

E³ 7

§1.3. Finite type Clos d Curves on Hyperbolic spaces 9

Chapter II. Gen ral Notation 11

§2.1. Pseudo- Riemannian Manifold 11

§2.2. Finite Type Subn1anifolds in Euclidean and Pseudo-

Euclidean Spaces 16

Chapt r III. Finite Type Curve 20

§3.1. Results on Finite Type Closed Curves in

Em, sm

^and

E;n

²⁰

§3.2. Th Classification Probl m of Finite Type Closed Curves

on Hm( -c²) 23

§3.3. Proofs of Theor m 3.5 3.6, 3.7 3.8 and 3.9 24 Chapter IV. The Finit Type Surface. of Revolution in E³ 33

§4.1. Th Cia ification Probl m of Finite Type Surfaces in E³ 33

§4.2. Th Classification of Slu·fac s of R volution I 34

§4.3. The Classification of Slu·fac s of Revolution II 42 Chapt r V. Topics Related to Finite Type Sub1nanifolds The-

ory

§5.1. Exampl s of Finite Type Slu·fac

§5.2. Related Topics

47 47 50

Typeset by A;\.;(5-'lEX

Chapter VI. Biharinonic Subn1anifolds 54

§6.1. General R suits on Biharinouic Submanifolds 54

§6.2. The classification Problcn1 of all Biharinonic Curves In

EF

⁵⁶

§6.3. The Non-Existence Th oren1s of Non-Harmonic Bihar-

monic Surfaces in

Et

⁵⁹

Chapter VII. Biharinouic Surfaces iu

E(

62

§7.1. Fundainental Forinulae on Biharn1onic Surfac sin

E(

62

§7.2. Biharmonic Surfaces with Nonzero Constant Mean Cur-

vature vector Fields in

E(

63

§7.3. Biharmonic Surfaces with Light-like Mean Curvature

Vector Fields in

E(

71

§7.4. Biharmonic W-Surfac s In

E(

77

Re£ r nc s 88

Pr face

In thi arti I we d al with th ubj t.:s rela.t l to two pr 11 m po. e l by B. Y. ,h n in [11]. One of th m i th la si£ ati n proll m of all finit. typ hyp r urfa.c sin En+l. Th oth r is th cla sifcation 1 r bl 111 of all biha.rn1oni sul 1nanifold in En+l.

The idea of submanifold of finit typ in a u lidean spac was ori inally introduced in [2] by Ch n. Aft. r th n, he d v loped it into a cone pt of finite typ ub1nanifold in a ps udo-Euclid an spac in [4]. To lay the finit. typ ubn1anifold th ory hav greatly d v lop d due to Ch n and hi collaborators ( e [3], [11 J tc.). In hort a u bmanifold of finite typ i d fin d as an i om trically in1m rs d n1anifold M in a Euclid an pa Em or a pseudo-Euclidiean pace E;n via an iso1netri i1nn1 r ion x which is expre sed a a finite sum of th igenvectors of the Laplacian ~ on A1 (see D finition 2.14 for th d tail).

Ac ording to thi d finition, th so-call d minimal urfaces are of null 1-type Euclid .an ph r s ar of 1-type and th ircular cylind rs ar of 2-type. On th oth r hand th llip oids, th paraboloids, the hyperboloids the tub except the ircula.r cylinder th non-flat tori

t .

are known as the surfac s of infinite typ . Th re also xist a lot of finit typ and infinit typ urface in a p eudo-Euclid an pac ([4] [6], [26], [27]).

Although the v ral partial solutions ha be n obtain d so far the cla ification prob- lem of all finit typ hypersurface in En+ ¹ i too difficult to b olv d on1pletely. For in tance, it i till open even to cla ~ify all finite type urface in a 3-diln n iona.l Euclid an pa.c E³. In fa. t in E³, no finite typ ~urfac .~ oth r than minin1a.l urfaces, circular ylind rs and ph r ar known o far. Ch n, Dillen Garay, V r tra. 1 n and Vran k n n id r d thi pr hl m about the urfac with certain re tricted conditions a th tube , th on th rul d urface th quadri urfa s in E³. Fro1n the re ult due to th m we know that th n1inimal urfa.ce the circular cylind r and the spher are the only finite typ urfa in la of uch urfa e in E³ ( ~ §4.1 ). In words of th differ ntial quation, th ir r ult how that n1ini1na.l urfac s circular ylinder or sphere are the only solutions of the lli ti typ cliff r ntial quation for th certain initial conditions. In chapter IV we will on id r thi typ of pro l m an1ong th . urfac of r olution in E³. Though it e ms that th surfac of r volution \\ill be the on iderabl target next to the abo e-m ntioned urfa es in E³ w do not hav u c d d o far in cla.s ifying all the finite typ urfac of r volution. W will m ntion about th partial result. obtain d so far in §4.2 and §4.3.

he clas ifi ation proble1n of 1-dinl n ional finit typ ubmanifolds nam ly finit type urv in a Euclid an pa or a Eu lid an ~pher were inv tigated in [3), [1 ], [19] etc.

Ac rdin to th pa. r v ry clo d 1-typ curve in a u lidean pa Em i pr is ly th cir l and th r ar 1na.ny k-typ urv in E³ for any k

=

2 3, · · ·. AI o only finit typ clo d urv in a 2-dim n ~iona.l ,ph r ² is a. ircle ( i . . , -type urv ) and only finit typ clo d curv in a 3-dinl n ional ~pher 8³ i a. cir l (i . . , 1- yp urv ) or a lo d W-curv ofra.nk 4

(i ..

2-typ urve). Ano1h r t.)p of pa fonn i th hyp rbolic pa . Th r for it i r ally natural to on i l r thi t.. pe of probl 111 for th cur on it.

vV

will inv sti at in chapter III th cia. ifica.t.ion probl 111 of finit typ lo d curves on th hyp rboli spa and 1-type I d cnrv ar on1pl t ly d t nnin d and th non-exist ncy of oth of2- and 3-typ clo d urv ~ on ff²(-c²) i aJ~o pro d.

The study of bihannonic submanifolds in a Euclidean pace was also started by Chen (see (11]). After then, I. Dimit.ric studied this subj ct and obtained several interesting results in his thesis at Michigan State University. In short, a biharmonic submanifold M is an isometrically irnmer eel Riernannian or ps udo-Riemannian rnanifold by an irnmersion x in a Euclidean space Em or a pseudo- Euclidean space Ern satisfying the differential equation ~ ²x

==

0 where ~ is the Laplacian on M. By the so-called Beltrami formula

~x

==

^-nH for an isometric immersion x of Af into Em or Et, the biharmonicity means that M is the subn1anifold of Em or E;n with a hannonic mean curvatur vector field H. Though the differential equation ~²:x:

==

0 has not such well-known and great physical background as the Euler-Lagrange equation ~:2:

=

0 in the harmonic (that is, minimal) submanifold theory, the physical idea of biha.nnonic subn1a.nifolds appears in the elasticity theory related to the clapped plate equation.

The real problem in the classification of all bihannonic submanifolds of Em is reduced to classify all non-harrnonic biharmonic subn1anifolcls in Em because every harmonic sub- manifold is biharmonic. However the inve tigation of the biharmonic submanifold theory is not so easy by the reason why the differential equation ~ ²x

==

0 is of the elliptic type and of order 4. In fact, in spite of great efforts, no exan1ple of non-harmonic biharmonic submanifold has been known so far. Neverthless the study is progressing step by step. We have known the certain bihannonic subn1anifolds satisfying the additional conditions are necessarily harmonic. For instance every biharmonic urface in a 3-dimensional Euclidean space is harmonic (Chen [11]). Every biharmonic su bmanifold with constant mean curva- ture is harmonic (Din1itric [32], [33]). Every biharmonic hypersurface with at most two distinct principal curvatures is hannonic (Dirnitric [32], (33]). Every biharmonic pseudo- umbili al su bmanifold of din1ension f- 4 is harn1onic ( Dirnitric [32], [33]). Every biharmonic curve is harmonic (Di1nitric [32], (33]). Based on all known inforn1ations Chen conjectures that he only biharmonic sub1nanifolcls in a En lidean space are harn1onic.

In contra t to th Euclidean case, w have a. lot of exarnples of non-harmonic biharmonic submanifolds in a pseudo-Euclidean space. Thus it co1nes up as our problem to classify all biharmonic submanifolds in a pseudo-Euclidean space. We will investigate in chapter VII th classification probl 111 of non-harn1onic bihannonic sub1nanifolds in a pseudo-Euclidean space. We will clas ify there all bihannoni curves in

E;n

and prove every biharmonic sur- face in a 3-dinl nsional pseudo- Euclidean space

Et

is harmonic. Moreover we construct son1e exa1nples of non- harmonic bihannonic surfaces in a 4-dimensional pseudo- Euclidean space

E(

and prove that a bihannonic urface with certain additional conditions is neces- sarily one of th e exa1nples.

Finally, I would like to expr ss my gratitu le to Professors Bang-Yen Chen, Hisao N a.ka- gawa and Katsuhiro Shioha1na for their friendly encouragement and support to my research activities throughout the preparations for this article. I also thank deeply my teachers : Professor Toshitan Uesugi and the lat Profes or Tsuneo Suguri for their mental support to me over many years.

Chapter I. In trod nc tion

In thi cha.pt r w will ou tlin th n1am a 1111 in thi a.rti l . In § 1.1 w introduc Ch n Conj ctur 1 and tunma.riz our result h or In~ 1 2 3 4 5 con rning it. In §1.2 we introduc h n s Conj ctur 2 and u1nn1ariz our re ults h or m 6,7, ,9 10 on rnin it. In §1. , w introduc on1 r ~ults a.bou t th finite typ curv in a uclid an pac or a uclid an pher and umn1anze our r ult h oren1 1112 13 14 15 cone rnin th finit typ ur in a h. p r oli space.

§1.1. Ch n s Conj ctur Cone ruing Bihar1nonic Snb1nanifold.

Th r arch of 1nini1nal urfac ~ in a 3-din1 n. ional Euclid an . 1 a E³ ha b n on of th 1110 t att.ra tiv an in11 orlant ubj ·ts among n1an · 111 t r and analy i t ~in

J.L.Lagrang . It ha b n on of ain1 in th inv tigation of 1ninin1al urface to find out n w xampl of n1inimal urfa e in E³. Through t.h int n ive ffort for it thi th ory has dev lop d into a furth r huge theory and built. a fruitful th ory f n1inimal u btnani- fold in a Ri mannian 1nanifold.

In thi arti l m tin1e call the 1ninin1al u tnanifold in Em the harmoni ubman- ifold in Em. h harmonic ub1nanifolcl and t.h iharn1onic ubmanifold in a u lid an

pa e Em or a p udo-Euclidean pac Ern are d fin d a. follow :

h har ni ub1nanifold i~ a Ri 1nannian or a p eudo-Riemannian 1nanifol M im- m r d into Em or E;:n by an i ornetric irn1n rsion x satisfying the differential qua Ion

~x

=

0 w h r ~ i th Laplac op rat. or on lvf.

Th iharm nic ubn1anifold i a Rien1a11nian or a p eudo-Ri 1nannian manifold M im- m r d int Em or E;:n by an i -·oi tric in11n r i n x a.ti fying the differ ntial quation

~²x

=

0. Wh n M i~ Ri n1annian ' all parti ularly it a pace-like harn1oni ub- manifold or a pa -lik bihannonic sub1nanifold. pa -lik harmonic subn1anifold is th o- all d 1ninin1al ub1nanifold. In a. 1Iink w~ki spac -ti1ne, it i usually call d to be maximal Fro1n th d finition , it follow m t.ant.ly that every harmonic ubmanifold is bihann n1 .

lth u h 1uat.ion ~ ² J_' 0 on th bihannonic ubn1anifold th ory hav n r at ph, ~i al ck ·r und in a tu r as t.h ~ ul r- La Tang quat ion ~x

=

0 on th minimal urfac ~ th ory, th th or.· f t.h biha.nnoni ub1nanifold is a.l o on of th atra ti ubj t in g om try. or in tan it i a n w and int re ting probl m to find out exampl of th biharmoni ubn1anifold into a uclid an spac Em or a p udo-Euclid an pac E;:n. in th B ltratni forn1u la ~x

=

-nff impli that ~ ²x

=

0 i qui al nt to

~H = 0 wh r H i th n1 an ur a ur tor fi ld of x th th ory of the bihannonic u bmani£ ld m to b ¥.

11

u ful to h ara t rize the eon1etry of the su b1nanifold with th harmonic 1n an urva.tur v tor fi ld a th hannoni au map of the in1m r ion into a pac f rm ha.ract riz th g om try of ~ubn1anifol [35) [46), [47), [5 ]). nd al o it link with th d ci ion probl m of th u 1 n1a.nifold with a pr crib d m an curvature vector ( [50]). More v r ~²x = 0 it If i a r. iffi ult but int r ting cliff r ntial

quati n in analy i too.

Now it is known that v ry hiharn1onic surfa<'(' in a. :3-din1ensiona.l EuclidE-an pa or a 3-dimen iona.l p. udo-Euclid an SJ a.c i.~ always han11onic. h n rov d thi. fa t for th surfac in a 3- limensional u li lean spac . J n ('h apt ('r \ 1 \\' a. s rt. that it i al o true for the surfac in a 3-diin n ional p euclo- Eucliclea n ·pa . That i we hav

Theor m 1. Let x be a biharn1onic i o1netric irnn1er ion of a . pace-like urface or a pseudo-Ri mannian surface into

Ef

^(t

=

1, 2). Tl1en xi l1armonic.

Mor ov r it i n1a.yb a natural que tion to a k wheth r thi i tru for a biharmonic ubmanifold in a high r din1 n iona.l Eucli l an or p. eudo- uclid an pa. . In fact, the following wa. propo d by h n.

Probl In 1 ([11]). Oth r tl1an harmonic. ubn1anifolds of Em, wl1ich ubmanifold of Em are biharmonic ?

To answer thi probl 111 i not o ea y. In fa t, not only the proof of non xi t ncy but also any xampl of non-harn1onic biha.nnonic ubn1anifolds in a.n rn(m

>

3)-diin n iona.l Euclid an pa e Em has not b n obtain o far. So1n r ults du to Dimitri ( [32] (33]) upport the non xi t ncy of non-harn1onic biharn1oni ubma.nifold in Em (s e chapter VI forth d tails). Ba. don known infonnations obtained so far, the following i conjectur d.

Ch n s Conj ctur 1 ([11]). Tl1e onl.Y biharn1onic ubmanifold in Euclidean spaces are harmonic on

In contra t to the Eu lidean ca e \\ hav a lot of example of non-harmonic biharmonic ubmanifold in a p udo-Eu lid an pace. \V will in e tigate in chapter VII a cla ification probl m of n n-harnlonic bihannonic ·u Inanifold. in a p eudo-Euclidean spa ⁰

vV

^will

con truct in §7.2 and §7.3 o1ne exa1npl of non-hannonic biharmonic surface in a 4- dim n i nal p~ u lo- uclid an ~pac

E[

and prov that every biharmonic urfac with certain additional ondition i n ce arily on of the e xampl s. That is we have th followin th or m (s chapt r VII forth notation in th m).

Th or m 2. Let M be a pace-Jik urface with nonzero con tant mean curvature in E[. Tl1en M i bil1armonic if and only if

(1) t

=

2 and, up to rigid n1otion of Ei, 1\[ i. a ._urfac giv n in Example 7.1 or

(2) t

=

1 and, up to rigid n1otions of R~¹, .\!

is

a surface gi" n in xarnpl 7.2.

Th or m 3. Let Jv! b a p udo- Ricn1annian urfac of ignature ( 1 1) with nonzero constant m an curvature and flat norn1al connection in E[. Then Af i biharmonic if and only if

(1) t = 3 and up to rigid motion of Ej Jt.f i a urface given in Example 7.3 or

(2) t

=

2 and, up to rigid motion of E:j, A1

i.

a. urface giv n in Example 7.4 or Example 7.5

or

( 3) t = 1 and, up to rigid motions of

Ef,

^{M is} a urface given in Example 7. 6.

Theorem 4. Let x : M ~

E(

be an i ometric in1mer. ion of a space-like surface (resp.

a pseudo- Riemannian urface of ignature ( 1, 1) ) Nf in to E(. Tl1en x is quasi-minimal and biharmonic if and only ift

=

1 (resp. 1

=

2 ) and up to rigid motions of

E{

(resp.

Ei ),

M is a surface given in Example 7. 7. ( r sp. Exa1nple 7. ) .

Theorem 5. Let x be a non-harmonic i ometric immersion of a spce-like W- u1-face M in E[. A sume that 3a²

+

E3G

#

0 for tl1e mean curvature a a11d the Gau curvature G of M. Then x is a biharmonic immersion with flat norn1al connection if and only if up to rigid motions of E(, M is a surface given in Exan1pl 7.1 a surface given in Example 7. 2 or a surface giv n in Example 7. 7.

§1.2. Ch n s Conj cture Concerning Finit Type Surfaces in E³

The id a of u b1nanifolds of finite type in a Eu lidean space was introduced by Ch n originally in [2]. Since then, Chen and his coil agu hav developed greatly thi th ory ( [3], [11] etc.). Before the appearance of paper [2] Takahashi obtained the following th orem (the so-called Takahashi's theor In) which i~ an interesting r sult about th 1-type

ubmanifold in a Euclidean sphere

sm

^{(in our word, th spherical 1-typ} sub1nanifold).

Theoren1 A ([59]). Let x be an i ometric immersion of a Riemannian manifold M into a Euclidean pace Em+l. If ~x = AX A# 0, then

(1) A> 0

(2) x(M)

c

m(1·), where Sm(r) is a hyperspl1ere of Em+! centered at origin and with radius r =

VnJX

and

(3) i minin1al.

Furtllermore, if X i a n1inin1al ll11111el'._ ion of ~I in to m, ( 1')' then ~X = rn2 X.

h r exi t a lot of finit typ surfar · in th fa1nily of th clas. ical surface in E³.

Minimal urfa , sph r ir ular cylind r~, flat tori etc. all are th urfa.ce of finit typ in E³. On th oth r hand, ellipsoid paral oloi Is, hyp rboloids, tub except circular cylind r , non-fiat tori tc. all ar kno\\ n a. th infinit type surfaces in E³ ( e [3] [7], [14], [15], [16] for xampl ) . It is also known that v ry non-harmonic biharmonic sub manifold in Em if it xi t , i of infinite type.

Th followin is an open proble1n propos d by Ch n.

Problem 2 ([11]). Clas ify all finite type ubmanifold in a Euclidean pace.

Thi problem ha b en inten iv ly inv st.iga.t d by n1any g om ter . However a lot of int restin part of it ar still OJ en. In particular it is very intere ting that the urface

of finit typ in a 3-dinl nsional Eu li l an pac kn wn o far ar only minimal urfa (null1-typ ), sph r (1-type) an circulcr cylin l~r (null 2-typ ). In ~pit of gr at fforts this very prin1itiv que t.ion of fin ing u t. any ot h r exan1ple has r n1ain I op n. For instan , in

[9]

hen ho\

l

a tuh xc pt a circular cylincl r i of infinite typ . In [37], Garay prov d that a on xcept a plan i: of infinit. typ . In [16) h n, Dill n V r tra l n and Vranck n show that a rul I ~urfa~ xc pt a plan a cir ular cylinder or a h licoid is of infinit typ . In

[14], h

ⁿ an l Dillen obtained that a quadric ex pt sphere or circular cylinder i of infinit typ .

Bas d on all known infonnations Ch n give a conJ ctur about thi pro l m a follows:

Ch n s Conjecture 2 ( [11]). Tl1e sph re i tl1 only compact urface of fin it type in a 3-dimen ional Euclidean pace.

In cha.pt r IV w will inve tigate the la ification proll m of all finite typ urfaces of r volution in E³ which 1nay 1 e one of the urfa s of next targ t to olve Probl m 2.

Though th urfaces of revolution in E³ i on of the simpl t surfa.c it In to b difficult t la ify then1 con1pl t ly. The followin Theorem 6, 7, 9 and 10 ar th partial an wers for thi probl m ( chapter IV for th not.a.ition ).

Theor m 6. Let M be a . urface of r volution of the polynomial kind. Tl1en M i a surface of finit typ if and only if 1\1 is either an open portion of a plan or an open portion of a circular cylind r.

Th or m 7. Let lvf be a urface of revolution of tl1e rational kind. Then M i a urface of finit typ if and only if M i an open portion of a plane.

In fa t h r In 7 f llow frotn th following 1nor g neral re ult:

Th or In 8. Let 1\1 be a finit type urface of revolution param trized by x(u, v) = (u o v usin v g(u)).

If g'(u)²

=

Q(7.t.)/R(u) for ome polynon1ial fun tion. Q(7.t.) and R(u) in u then M i an open portion of a plan , or M i an op n portion of a catenoid or deg Q

=

d g R

=

2

+

^{deg (}

^Q +

R).

Th or n1 9. Let M b a urface of r volution in E³ defin d by

· ( u. v) = (

f (

1.1.) o v

f (

1.1) ~j n v g ( u))

where f('u)

>

0 i non-con tant and 'U i th arcl ngth of .·z-plane curve (f(u),g(u)). If f(7.t) i a r ratively prime rational function uch that f(u) -::/= u or f

=

P/Q for ome polynomial functions P and

Q

in 7.l of de ( P) -::/= de ·( Q) then A1 i of infinite type.

Theor m 10. Let M be a urfac of r volu tion in E³ defin d by x(u,v)

=

(f(u)co v,f(u) inv g(u))

where f(u) = P(u)/Q(u)

>

0 i a non-con tant relatively prin1e rational function with d g(P)

<

d g(Q) and u(s) is a function of where s i. tl1e arcl ngth of xz-plane curve (f(u),g(u)) such tl1at u'(s)² and u"(s) are polynomial functions in u of degr e 2 and 1 respectiv ly, then M is of infinite type.

§1.3. Fiuit Typ Clo d Cnrv .s on Hyperbolic Spaces

Finit typ curv 1n a uclidean spa e ^w re inv ~tigat d in [3)

[1 ], [19)

^etc. For instance,

Th or m

B ([3]).

Let 1 be a closed plane curve of finite type. Then 1 i of 1-type and hence 1 i a circle.

In [3), Chen al o proved that every closed curve of k-type in Em lies in a sub pa of dimension n~ with m :::; 2k. Com ining thi fact with Theor m B, th following i obtain d too.

Th or m C ([2],[3]). Every clo ed 1-type curve in a Euclidean pace Em i pr ci ely the circle.

In

[19)

h n Dill n and V r traelen cla ifi d all clos d 2-typ curv s and prov d that th r ar many k-type curve in E³ for any k = 2, 3 · · ·.

h f ll wing r ult ar relat d to th sph riral finite typ Io~ d curve ( §3.1 for th notati n ) .

Th or m

D ([1 ]).

Only clo. d finite type cun e in a 2-dimen ional phere is the circle.

Th or m E ( [1 ]). Only clo ed fin it type curve in a 3-dimen ional sphere is the circle or the clo d W -curve of rank 4.

We will inv tigate in chapt r III th fin it type clos d urve on a hyp rbolic pac r aliz d in a Minkowski pa -tim . We d not by

a .

0 the vector d fin d by

for a v tor ao

= (

^{a.1 a.2, · · · at · · ·} am) in E;n. very clo ed urv 1 : [0, 21rr) ---+ EF of th length 21rr in

E;n

may b r arded a a.n is01n tric in1n1er ion of a circle 5¹ ( r) of radius

1· int E;n. W u th arclength

s

a a. paran1 t r of I· Th n th Laplacian ~ on 5¹ (

r)

is given by~= -d² /ds² and th i · n\alu ar {(//r)²; l = 1 2 · · · }. Th corr pondin

eigenspa Vii construct d by u ing co·(ls/1') and ·in(ls/r). II nc ev ry clo d curv 1:

[0,

27r7'] --+

Er;n

^{ha th p ctral} decon1po. it ion

00

1(s)

=

^ao

+ L

^{!t(s) 11}

⁼

^{a1 c (l /1')}

⁺

^{b1 in(l /r)}

1=1

where a1, bz ar orne v ctors in Er;n(s [3],[1 ]).

We prove in chapter III the following th or m cone rning 1-typ clo d urve on a hyperboli pace Hm( -c²) in a ~1inkowski pa tin1e (s chapter III forth notation ).

Th or m 11. Only 1-type clo. d cu1·v on !Jw ( -c²⁾ i. an inter ction of Hm( -c²⁾ and som 2-pla n P lying in ITao. II re

n

ao denote a hyperplan through a₀ '.vl1ich i.

orthogonal to the vector

a

0 in tl1 ensc of Euclidean scalar product.

The following is a non xi t nc theor n1 of ~·-typ (k

=

2 3) clo l curve on H²( -c^2).

Th or n1 12. Ther exi t neither 2-type clo. eel curve nor 3-type clo ed curv on H2( -c2).

For a high r k-typ ( k

>

^{2) clo ed curv} atisfyin a certain additional a u1nption on H²⁽ -c²), w hav orne results. For xan1ple

Th or m 13. There exi t no 6-type clo d curve 1(s) with

<

^{1(1)( )} 1(^{1)( )}

>=

con tant (/ = 2 3) on H²

(-c

²).

Th or m 14. Ther exi t no 5-typ lo ed curve 1(s) l-vith

<

¹⁽²)(s), /(²)(s)

>=

con tant (i . . witl1 con tant cu1Tatur ) on H²( -c²).

Thor 111 15. TJ1 r exLt no 4-type lased ^Ul'\' !(._) witl1

< /(

²) ( ) /(2) ( )

>=

con tant (i . . \Vith on tant curvatur ) on f/²( -c²).

R mark. R _ ntly h n ha 1nv tiga.ted a. das ification proble1n concerning 2- or 3-type ub anifold in th hyp r olic pace in1b ld d standa.rdly in the Minkow ki pace- tim . For xampl , h proved

Th or n1 F ([13]). Ther xi t no con1pa t :.--type hyper urface in the l1yperbolic pace in1b dd d tandardly in th Aiinkow ki pac -tin1e.

Th or In G ([13]). Ther xi. t no 3-typ l1yper urface with con tant m an curvature in the l1yperbolic pace imbedd cl tandardly in tl1 Afinkow. ki pace-time.

A part of Th or m 12 i includ din Th oren1 F and Th or 111 G. Th or m 12 al o how that for th los d 3-typ curve on hyp rboli 1 a of dim n ion 2, w ca.n r rn v the condition of c n tant m an urvatur fr01n Theor 111 F.

Chapt r II. G n ral Notation

In the following, we a un1e th ubn1anifol l. ar all conn t d and of po itiv din1 n 10n.

Mor over th cliff r ntiability of geon1et.ri obj ts i a ~tuned to b always of ⁰⁰ -cl . In

§2.1, th univ r al notations and notion in th 1 ~ ud -Ri ma.nnian g 01n try ar d fin d and som fundam nta.l forn1tda u. ~ l in this art.i l are dis u d. In §2.2, w introduc th id as of th biharn1oni ul n1a.nifold. and t.h finit. type. ubn1a.nifold. in a uclid an

pace or a p. udo- uclid an pac .

§2.1. P udo-Riemannian Manifolds

We will surv y briefly in thi ection th funda1nenta.l con pt and facts in th p ucla- Riemannian g o1n try for later u . We ^r f r n1a.inly to 0' il ([55]) and Ch n

([4],

[6]).

For th g n ral one pt in th Rien1annian g 0111 try, r f r to th book of Kobay hi and omizu ([51]).

Let M be a C⁰⁰-cla cliff rentiaLl n1anifold of lin1 n ion n and g a C⁰⁰-cla differ n- tiable ymm tri nondegen rat t n or fi ld of typ ( 0 2) on Af. Th gp at v ry point p of M d fin th scalar product on the tang nt pa. e Tp(M) of M at p. Th ind x of gp is not nece arily onstant in general.

Definition 2.1. (1) If the ind x of gp i con tant t(O ::;

t ::;

n) on M w all g a p eudo-Ri 1nannian metric of signature (t n- t).

(2) A C⁰⁰-cla differ ntiabl Inanifold ( I, g) furni h with a ps udo-Ri Inannian ^lll t- ri g i all a p u do- R i 1n an n ian m ani

f

o I d.

(3) A p. udo-Ri 1nannian n1anifold of ~ignaturc (1, n-1) i~ a. ~a- all d Lor ntz manifold.

A p udo- .i mannian 1nanifold of si ,·natur (0, n) n1ean a Rien1annia.n 1nanifold.

W r mark tha in th book [55) of 0 il a p~ udo-Ri n1annian 1na.nifold i called a mi-Ri mann ian manifold. he in1pl ~t xa.n1pl of p eu a-Riemannian manifold i a p eud - u li an pac d find a follow. L t (a.~¹ x², · · · , m) b a point in Rm. For

ach

t

(0::;

t::;

m) w d fin a calar produ t go on Tp(Rm) at the point p of Rmby

t m

go(vp wp) = - L ·liwi

+

L viwi

i=l i=t+l

wher ^Vp

= L:;:

₁ ^{vifJ/8 i and Wp}

= L:;:

₁ wi()j()xi.

E;n

denote a p eudo-Riemannian n1anifol Rm with a anoni alps udo-Ri 1nannian n1 tric go.

Definition 2.2. (1) g₀ i call d a p eudo-Euclidea.n metTic of ignature

(t

1n- t) and

Er

^{i all a} p udo-Euclid an pac of ignatur (t n1- t).

(2) E! i th a-call d J..finkow ki pac -tim . . E

0

^{111 ans a. uclid an ~pa Em.}

From now on w will on1eti1n of go.

Definition 2.3. (1) :n(a., 1·)

= {p

E E;n+l

I <

^p- a., p- a

>=

^{1·2} call}

d

a p u do- ph T f r a l i u ^7'

>

0 and cent r a in '

;n

^{+ 1}

(2) H;n( , r) = {p E E;~i¹

I <

p-a, p- a>= - r²} i call lap udo-hyp rbolir pace of radiu r

> 0

and ent r a in E;~t¹.

In th f llowing, w will sinlply d not b. .

-In (

7') th p ucl - ph ere

s;n (

0 r) and by

H;n (

^{- r2)} th p udo- hyp rboli pac

Hr

^{1 (}

⁰

^{r) r} p tiv ly wh r

0

i th origin of

Ern.

sr(r) is th o- alled d Sitt r pace-tim and H!( - r²) (or th uni r al OV rin of it) is the so-call d (or re p. univer a~ anti-d Sitt r pace-ti1ne. W r mark that 'F( r) is cliff omorphic to Rt ^X sm-t and Htm( ^{-7· 2 )} t t X rrm-t.

Hm( - r²)

=

{p E E~+l

I <

p p

>=

- r² P1

> 0}

i~ call d a t.andardly i1nb dd d hyperboli pace in the Minkow ki spac -tin1

E't.

It i a on1pl te Ri mannian manifold of con tant urvatur

-1/

^1·2.

On th p udo-Riemannian n1anifold w an a.l. o dis u th g om t.ric cone pt i1nilar to tho on th Ri n1annian n1anifold . uch as the Levi- ivita connfction, t.h para// I translation, th g odesic curve, the fXponcutial map, the Riemannian C7trvatur t nsor th Ricci curvatur tens07' th calar curvature, th Laplacian and o forth. hou h we n d om tim giv the ar ful at.t ntion to do it. their cl finit.ion ar aln1o t parall l to on in Ri mann ian ca. ( [55) for t.h details).

Definition 2.4. Let v b a tang nt ctor to a p udo-Ri mannian manifold A1 with a pseudo-Ri mannian metric g.

(1) v 1 aid to b pace-lik if g(v v)

>

0 or v

=

^0,

(2) v 1 aid to be light-lik if g(v 1) = 0 an v

#

0, (3) v 1 aid to b ti1n -lik if g(v v)

<

0.

Definition 2.5. ( 1) A las { 1 , 2 · · · n} of th tangent vector fields d fin d on an open u t U of a p ud -Ri mannian n1anifold (A1,g) i called an orthonormal y tem on U if

g(i i)=Ei=±1 g(i j)=O (i#j)·ij=12··· n.

(2) Wh n it mor ov r a i n to

w call it a local frame

fi

ld of A1.

ry point. p in U a ba i ~ of th tangent pa Tp ( M)

L t (M g) b a p udo-Ri n1annia.n manifoll with a ps udo-Ri mannian metric g and

{ e1 e2 · · ·

m}

a lo al fran1 fi lcl of 1'!. For an enclon1orphis1n F of th tang nt bundle T(M) of 1\rf w d not y lra.c F th t.ra c ofF i.e.,

111

l1·ac F =

L

^{ig(F i ei ).}

i=l

Th gradi nt grad (f) of a fun tion

f

on Af i ~ 1 fin cl as a vector fi ld on M ati fying g(grad (f), 4\) = X f

for ev ry tang nt vector fi ld to A1.

For a tan n t v ct or fi ld 1~'" to A1, t h d i r n e d i v ( Y) of Y n A1 i d fin d by

div (Y) =

L

^{Eig(\l ; Y ei)}

i=l

where \7 is the Levi-Civita connection of A1.

These definitions are independent on the choice of local frame field of M and hence they are well-defined functions or vector fields on A1.

Definition 2.6. For very sn1ooth function f on M, we define a smooth function !::l.f on M by

!::l.f

=

-div(grad (f)).

/:). is the Laplace operator or Laplacian of A1.

From definition, we see, usin , a local fran1e field { e_{1 ,} e2, · · · , em} of Jt.1,

m

!::l.f =

-LEi{

eieif - (\7 ^e, ei)f}.

i=l

!::l.f can be al o expressed as follows;

/:)., = - 1

~~( ij ~)

f Jl

^det(g;i

)I i~l

Ox' g fJxi ,

where (gii)

=

(gij) ^{- l} is the inverse of matrix (9ii) associated with the pseudo- Rie1nannian metric g.

We define the Laplacian !::l. for every E;n-valued smooth function f

=

(fl, !2, · · · , fm) on M by

Definition 2. 7. Let J..![ and M be two pseuclo-Rie1nannian m!l.nifolds with metric ten- sors g and

g

r spectively. An is01netric immersion fro111 M into J..1 is a sn1ooth immersion

f: M ---+if with f*(g) =g. Then J..1 is said to be a pseudo-Riem.annian sub1nanijold or simply a subn~anifo/d of

A1.

In particular, when P.1 is a R.ien1annia.n su bn1an ifold of A~!, it is called a space-like sub- manifold.

In the follow in , we will treat a. Ri n1an nian n1anifold also as one of a pseudo- Rien1annia.n manifold if without mentioned specifically.

Let M be an isometrically inunersed subn1anifold of a pseudo-Riemannian manifold

M

and \7 and

\7

denote the Levi-Civita connections on M and ii[ respectively. We denote m

=

dimM and n

=

di1nM. Then for every vector fields X,Y on M, the Gauss form:ula is given by

VxY· =

\7xY

+

h(X, Y), where h is the second fundan1ental forn1 of A1 in

M.

We denote by D the normal connection on th nonnal bundle Tj_ ( ^{A1) of M in}

M.

^Then

the Weingarten fonnula is

for every normal vector field ~ and for v ry tang nt v ct r fi I X to M. H r Ac i th Weingart n map deriv d from ~ whi h i r lat. cl with by

g(h(.X, Y) ~)

=

g(Ac~\, Y).

R mark. (1) vVhen M i a Ri 1nannian n1anifold, Ac a si ·n to ev ry point p of M a self adjoint ndomorphisn1 of TP (Jvf). Hen Ac i dia.gonalizable.

(2) Wh n M is a. p udo-Ri 1nannian 1na.nifoll, we r mark that Ac cannot b diago- nalized in g n ral

([53]).

Th Ri mannian urvatur t n or fi ld Ron a p udo-Ri mannian manifold M i d fin d by

R(X Y)Z

= (\7x

\7y]Z- \7[x,Y]Z

for all v tor fi lei X, Y, Z on A1. \Ve will d not by R and

R

the Ri n1annian urvatur ten or fi ld. of A1 and 1\t! r p tiv ly. ThC'n t h Ga1tss equation i ·iven b.

(2.1) g(R(.\ Y)Z vV)

=

^{[;(RC\, Y)Z} HI)+ g(h( .. \ , H ) h(Y Z)) - g(h( .. \ Z) h(Y H'))

for all v ctor fi ld . .. X, Y, Z, Won M. Wed fine the covariant d rivativ

V

x h of th cond fundamental forn1 h by

(Vx h)(Y Z) = Dx h(l', Z)- h('J x Y Z)- h(Y

\7

x Z)

for all tan nt v tor fi ld ~Y, Y Z to A1. Th n for all tangent v ctor fields X Y, Z to Arf th Codazzi quation i giv n by

(2.2) (R(}:_ l'')Z)l.

=

(Vxh)(l'' Z)- (Vyh)(X Z), wher (R(.},: Y)Z)l. i th nonnal on1pon nt. f R(~\ 1 )Z.

Th normal urvatur t n or field RD associat d with th normal conn ction D of M in

M

i d fin d by

RD (~\, Y)~

=

[Dx, Dy ]~- D[x,Y]~

for all tang nt ctor fi ll ~Y Y and for all normal ,. ctor field ~ to Af. Th n th Ricci q11 ation i n by

(2.3)

for all tan nt v tor field ~ ,, Y an l for all norn1aJ v tor fi Ids ~ and 17 to M. If th ambient pac

M

is a pac of con ta.nt ( tiona.l) urvatur k t.h n the quation of Gau s, oda.zzi an Ricci r du re p ctiv J. to

(2.4) g ( R( X, Y) Z W)

=

^k

{g

(X Ttl!)

g (

1'' Z) -

g ( )(

Z)

g (

Y Ttl!)}

+

^{g(h( r,} W), h(Y, Z))- g(h( .. \, Z) h(1⁷ W)) (V X h)(Y, Z) = (Vy h)( .. \, Z),

g(RD (X 1⁷)~, 17) = g([Ac A1

7

^{](~Y) Y).}

For ev ry nonnal v tor fi ll ~' w defin the covariant. derivativ \7 A~ of th W ingarten map A~ 1 y

wher X and Y ar th tang nt v ctor fidds to H. Then w hav g( (\7 X A~ )Y, Z) -?;(AD X~ Y, Z)

=

g( (V X h)(Y, Z), ~) for all tang nt v tor fi ld ~Y Y1 Z and for a 11 norn1al v ctor fi ld ~ to Jt.1.

W hoo a local fram fi ld { e1, 21 • • • , n n+11 • • • em} adapted to th p ucla- Riemannian m tric

g

of

M

in uch a way that, r stricted on th p eudo-Riemannian ub- manifold M { ^1, e2, · · · en} are tang nt to Jt.;J. w . that { e1, 2, · · · , en} i a lo al fra1n fi ld adapt to th indue d ps udo- Rien1annian m tric g on M and { en+11 n+21 • • • , m}

is a local fi ld of orthonormal fram TP rP(Jt.f) in A~!. From now on w put < A A

>=

EA

=

±l(A

=

1,2, · ·· n~). L t {w¹,w², · · · 1Wm} b th local fram field of dual1-forms of { 1, 2 · · · m}· The conn ction forn1 w~ i~ th n iven by

rn

d A

= L

^{w~ B · w~}

⁼

^{-EA c nw~, A B, C}

⁼

^{1, 2 · · · 1m ..}

B=1

From thi it foll w

HI

dw~ = - L

^{w A /\ w3.}

C=1

Put

h

= L

^{hrj w i wi T : i 1 j}

⁼

1 2 . . . , n 1'

=

^{1' 2 ...} m'

i,j, r

wh re h i th cond funda1nental fonn of AI in A!. h n

n

(2.4)

wr ⁼ L

^{hrj wi :}

^<

^{A r j}

^>=

^Er

^h~j.

j=1

D fiuitiou 2.8. Th m an curvatur v cloT field H of AI in if i a normal vector fi ld toM d fin d y

1 1 ^II

II= -t1·ar h = -

L

^{;h( i).}

11 II

i=l

D finition 2.9. L t Jt.;J be a ubn1anifoll [if. Th n

(1) M i all d to b a 1nini1nal or hannonic ubm.anifold of

M

if H vani h id nti ally, (2) J\1[ i all d to b a quasi-minin~a/ snbm.anifold of A~I if < H, H

>=

0,

(3) lvf i all d to be a pseudo-ttm.bilical ttbm.anifold of if if< H, H

>#

0 and Ay =a!

for a c rtain function a on M wh r I is th i 1 ntit.y tran formation of T(A1).

§2.2. Finite Type Submanifolds in Euclid an and Pseudo-Euclidean Spaces Let M be a pseudo-Riernannian n1anifold of din1ension n and ~ the Laplace op rator on M. We will prepare the fundarnenta 1 r suits on th finite typ th ory.

Definition 2.10. If a srnooth function

f

^on 1\1 is a solution of the differential quation

!::,.f

=

).j

for a certain constant A, then ). is an eigenvarue of ~ and

f

^an eigenfunction of /::,. cor- responding to A. If, in particular, f sa.tisfie::; D.f = 0 then f is a harmonic function on M.

Definition 2.11. A smooth function

f

on a pseuclo-Riernannian manifold A1 is said to be of finite type if it can be expressed as a fin it sun1 of the eigenfunctions of/::,., Otherwise

f

is said to be of infinite type.

It is well known that if lvf is con1pact, th n the Laplace operator ~ of 111 has the infinite sequence of eigenvalues such that

The dimension of each eigenspace Vi~ of ~ corresponding to the eigenvalue ).k is finite and V0 is 1-dimensional space consisting of th constant functions on M. It is also well known that if M is non-cornpact then A's are not necessarily positive, dim Vk is not necessarily finite and the eigenspace V₀ has the dirnension

2:

1 in general.

The volume element of an m-din1ensional pseudo-Riemannian manifold M is defined as a smooth rn-form w on M such that w( e1, e2, · · · , em)

=

^{±1 for every} local frame field

{ e1, e2 · · ·

m}

on M. In local expression, the volume element w is

As usual, we denote w by dfJ. vVe define an inner produ t (f_{1 ,}

f

_{2 )} for the smooth functions

!1, !2

^{on a} cor 1pa.ct rnanifold M by

Then th d cornposition 2::~

1

^{Vk i} orthogonal and dense in the space

coo (

^{1\1) of all}

smooth functions on M. Hence each sn1ooth function

f

has th spectral decomposition in

00

f = L

^{!k ; fk E 1/k·}

k=O

Let x : M ~ E;n be an ison1etric in1111ersion of a pseudo-Rie1nannian n1anifold M of dimension n into an m-dirnensional pseudo-Euclidean space E;n of signature

(t

m -

t).

Each A-th coordinate function x A (A

=

1, 2, · · · , 1n) of x

= (

x1, x2 · · · , Xm) is expressed as

QA

XA

= L

^{(xA)k : (xA)k E Vk.}

k=PA

If each A-th coordinate function xA(A

=

1,2, ... w) of:1·

=

(x1,x2 ... ,xm) is offinite type, then PA and qA in the expression above of :t:A are t.h nonnegativ int g rs satisfying

PA

:s;

qA. Ifsome A-th coordinat function .r.tt(A

=

^{l,2,··· ,n1) of;r}

=

(x1,:z·2 ··· ,xm) is not of finit typ , then in the expression al>ov of ;r A p is an inte "er and qA

=

oo. If we put p

=

^minA

^{PA}

^{and q}

=

ma.xA{qA}, then pis an int. ger an l q an integer or oo satisfying p

:s;

q.

Fork such that (xA)k =f. 0 for some A, we denote Xk

=

((xt)k, (x2)k, · · · , (xm)k). Then

xk is a non zero map of Minto

E'F

satisfying ~xk = ).k:z:k.

Definition 2.12. A E;n-valu d sn1ooth function

f

^{on a} pseudo-Riemannian manifold M is called to be harmonic if and only if it. atisfies the differential equation ~~ = 0.

In particular, if x is an isometric in1m~rsion of a pseudo-Riemannian manifold M into a pseudo-Euclidean space E;n satisfying ~x

=

0, then M is called harmonic submanifo/d and x harmonic immersion.

Definition 2.13. Let x :M_,. E~¹¹ be ani on1 tric imtnersion of a pseudo-Riemannian manifold of dimension n into an m-ditnensiona.l pseudo- Euclidean space E;n of ignature (t,m-t).

( 1) If x has the spectral decon1position a. follow:

k

(2.5) _X

=

X 0

+ L

^{Xi t :}

^{~ l~}

^it

^{= )..}

^{i t J' i t , ;z· it}

^#

^0'

^J~

^it

^{=I :l~}

ⁱ

^~

^(it

^#

^·i

^~

^{) .}

t=l

then M i called to be of k-type subn1anifold and x k-type immersion.

(2) If one of ).i₁, j

=

1 2 · · · , k in the d con1position (2.5) is zero, then submanifold M or immersion x is said to be of null k-type.

Remark. her exists no co1npact null k-t. pe subtnanifold.

Definition 2.14. Let x : M _. Eln be an ison1 tric in1mersion of a pseudo-Riemannian manifold Minto a pseudo-Euclid an space

E'F

of signature

(t,

m-

t).

(1) We all M or :r to be of finite type if A1 is of k-type for some positive integer k.

(2) M or x is said to be of infinite type if it i not of finite type.

We can say n1ore infonnation about a. cotnpact space-like submanifold Af in

Er.

^First,

as mentioned above each ).i

1 in the spectral decotnposition (2.5) is nonnegative. Secondly xo in (2.5) is onstant and in fact it. is th c ntroi] of ~1, that is, we have

:z:o = uol( 1 \f)

.l

M :z·d(J'

where vol ( 1111) i th vohu11 of A!. Also th sp ct. ral lecotnpo ition ( 2. 5) of x is orthogonal that is , w have

(xi₁^,Xi₁)= (

go(l~i 1 ^{,Xi 1} )d<J'=0

^(j:fl).

J .M

The following theorem is well-known a.ncl usefull in this article.

Theor m 2.1 ([3],[4]). Let x be ani. cHnc>t.ric in1n1crsion of an n-din1ensional pseudo- Riemannian n1anifold M into a ps urlo-Euclidcan spncc> L';n. Thf'n n· lJa,·e tl1 . a-call d Beltrami forn1ula

~~·=-nil,

where H is th mean curvature vector fi ld f :r.

Definition 2.15. L t x : M - E;n b an i~onl tri imn1 r ion of a p udo-Ri n1annian manifold lvi into a p eudo-Euclid an space E;n of ignatur (t, m-t). If x ati fies th dif- ferential quation ~²x

=

0, th n !vii~ ~aid to 1 biharmonic ubmanifold and x biharmonic

. .

1mmer 1 n.

Remark. ( 1) Th re xists no co1npact bihannoni space-like ubmanifold in E;n.

(2) Ev ry non-harmoni biharmonic spa.c -like subn1anifold in E;n i of infinit type.

The simpl t ubmanifold next to the hannoni on in E;n i of null 2-typ in E;n. Th r ar som result concerning th u bn1anifolcl of t.hi t.yp . In [9], Ch n cla ifi d th null 2-typ urfac in a 3-dimensional Eu lid an ~pac E^3. In (44], Houh cla ~jfi d th null 2-typ urfac s in a 3-diln n ional !\1inkowski pac -tim

Ef.

The cla . ifica.tion probl m of null 2-typ pa -lik tufa in a ti-din1 nsional ps udo-Euclid a.n 1 ac

E[ t =

1 2 w r al o eli u ~ d b. h nand on · ([26) [27)).

Now th following theor 111 pr nt~ a n ~ ·ary on lition for At! to b of k-type ub- mani£ ld.

Th or m

2.3 ([3] [4]).

L t A1 be a p. eudo-Rien1annian ubmanifold of E;n and H th m an curvatur ector fi ld of A1. If ~I i of k-type, tl1 n there i a polynon1ial P(X) of degr k uch that P(~)H = 0.

If M i ompa t, th onv r aJ~o at.i ·fied. That is

Th or m 2.4 ([4]). Let A1 be a con1pact p eu lo-Rien1annian ubmanifold i ometri- cally immers d in a p udo-Euclidean pace E;n. If there exi t a nontrivial polynomial P(X) uch that P(~)H

=

0, then AI i of finite typ .

u h a polyn01nial P(~\) in Th oren1 2.:3 is call(' I t.h m.inimal polynomial of finit typ ubmanifoll. Let Jt.I h of !t·-t.yp and (:...5) t.h(' ~pect.ral de o1npo it.ion of x. Th n th minimal polynon1ial P(.:'\) of AJ an he un iquly det. 'nni ned. for pr ci ly P( ..tY) i IV n by

k k

(2.6) P(X)

= L(

^{-l)i i..{yJ.~-i}

⁼ IT

^{(X- Ai})}

i=O i=l

where

c2

= L

^{).ill ).iJ2'}

j 1 <j 2

Ck

=

).it).i2 . . . Ai~;·

In contrast to con1pa.ct a , the xistence of a nontrivial polynon1ial P(~\) for non-con1pact submanifoll lvf u h that P(~)H

=

0 dos not n ^\C' ·arily in1ply in gen ral that M is of finite type. How v r, th conv rs of Theore1n 2.3 i ~ true for c rta.in sp cial non- on1pact p eudo- Riemannian su bmanifolds, as tat d

Theore1n 2.5 ([22]). Let x be an isometric irnn1e1·sion of a pseudo-Riemannian ub- manifold of a p ucla-Euclidean space E;n.

( 1) If dim M = 1 and there exi ts a polynon1ial P( X) satisfying the condition P( ~) ( x- c)

=

0 for some con tant vector c in E;n, then !11 i. of fi11ite type.

(2) If dim !11 ~ 2 and there exists a polynon1ial P(.\) of degree k with exactly k di tinct real roots sati. fying the condition P(~)(x- c)= 0 fo1· ome constant vector cinE;:, then M is of finite type.

Finally w d fin inductively a n1a.p Yl ( l

=

^{1, 2} · · · ) of Jt.;f into E;: by

for om

Y1

=

~x - -\1 (~:- ·a), Y1 = ~Yl-1 - AiYl-1

n t.a.nt. A[. Th n th follow in is t.ri \ ial.

Propo ition. Let x be an i on1 tric in1n1eLion of a p eudo-Riemannian ubmanifold of a pseudo-Eu lidean pace E;n. Tl1en

(1) x i of k-type if and only if Yk

=

0 and Yl

'/=

O(l

=

1, 2, · · · , k- 1) for some set {-\1, -\2, · · · , ).k} of the real nu1nber.

(2)

x i of infinite type if and only if Yk

i= 0

for any k and any set { ).1 -\2 · · · ).k} of the real numb rs.

Chapt r III. Fiui t Typ Closed Cnrv s

Th purpo of thi chapt r i~ to d scrih some known r suit.· ncerning th finit type curv son a hyp rbolic spa and prove 11Horeins :3.5, :3.G, 3.7, 3. an l 3.9. W r f_r to [3), [1 ], an l [19] for veral xan1ple. and restdts concPrning th finit type curve in a Euclid an an a 1 seuclo-Eucliclean pace. In §:3.2, W(' r sume our results and prove th m in §3.3. §3.1 i a ign d to th r vi w of t.h finite type curv in a Euclid an pa , a Euclid an ph r or a p u lo- Euclid an spac .

§3.1. R suits on Finit Type Clo. ed Curv s in Em, sm and E-;n

Vle begin thi chapter to r vi w s veral known results one rning the finit typ curv in a Eu lid an pa Em and a Eu liJ an sph re c;m.

Exampl 3.1. Th circle x( )

=

7'( os(sj1·) sin(s/r)) in E² i 1-typ curve.

B cau of D.= -d² /ds² forth circl ~:(-"), w hav D.x = x. ¹

As th r suits r lat. d to Exa1nple 3.1, th followin have b en known.

Theor m 3.1 ([3]) L t 1 b a E1Lclidean-planr curve of .finit type. Then 1 i of1-type or harmonic. H nc 1 i a part of a ci1·cle or a 8/rnight lin .

Th or In 3.2 ([18]) L t 1 b a clos d rurvf. in Em. Then 1 i of 1-type if and only if it i a eire/ .

W all a urv 1 in Em to b ph rica/ if it ha ~ th in1ag in a Euclid an sph r in Em.

A the r ult r lated t ph ri al finit typ urv , th following hav b n known.

3.3 ( [18]). A circle i the only clo d curv of finite type in a Euclidean

Th or m 3.4 ( [18]). L t 1 be a clo d curve of finite typ in a 3-dimen ional ph ere

5

^3. Then 1 i of 1- or 2-typ .

Remark. Th re ar Inany xa.n1pl ~ of non-sph rica! finit typ clo d curve 1n a Euclid an pa.c . For instan ,

Exa1npl 3.2 ( [3] ). A clo d curve in £³

x( )

= (

-3 sin(s/G) + c >.·(:;/2), -:~co.(._ /G)+ ~in(s/2), (3/v'2)(cos(s/3)

+

sin(sj:3)))

is of non- ph ri al 3-typ .