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(1)ON ISOMETRIC IMMERSIONS OF Si-INVARIANr]] MANIFOLDS Masao. Maeda Faculty of Education and HumaB Sciences Department of Mathematics 1fokohama Nati'onal University'. 1. Introduction. Riemannian manifold M is called to be Si-invariant if its isometry grgup I(M) containes the circle Si as a subgroup. Properties of Si-invariant Riemannian manlfolds are found in [1]. When such M is diffeomorphic to the 2-dimensional sphere S2, isometric imbeddability of M as a surface of revolution in 3-dlmensienal Euclidean space E3 is treated also in [1]. Using these results, M.Engman gave anether necessary and suMcient condition that M can be imbedded isometrically as a su'rface bf re'vlolution of class Ci in E3 in [21. Imbedding of M as a･ surface of revolution in E3 is unique if the rotati'on axis and the initial points are designated. Further-more, smoothness of class Ci of this imbedding c'an not expect to give smoothnes.s of class C2 in genaral. Mally imbeddings are riot Qf class C2. Now the purpose of this note is to give an immersion of certain Si-i･nvariant manifold ih. E3 which. looks like a surface ef revolution and have C2.smoothnessl' Thls immersion coinside with a surface of revolution if this immersion satisfy certain conditions.. 2 Si-invariantmanifolds Let M be a Si-invariant 'Rjemannian manifold diffeomb'rphic to S2 'and ･g the Riemann'ian metr'ic on M which is of class COO. In M, there exist two poi'nts pi and p2 whieh are fixed by Si-action 'of isomtry. group I(M). Let U denote the Qpen subs.et M - {pi,p2} of M. Then the Riemannian metric g on M is expressed as. ' ds2+a2(s)de2･･ g= ' coordinate on U around pi such that s is the-arc-length where (s,e);o < s < L,o S e < 2r is a polar of alI geodesics from pi to p2. a: [O,L] - R is a smooth function which is the solition of the Jacobi. . .･ a"+K('s)a(s)=O equation ･ ' .'. ' tt. with initial conditions a(O) = a(L) = O and a'(O) = -a'(L) = 1 where K(s) is the Gaussian curvature of M at (s,e). Since M is Si-invariantj a(s) extends smoothly on (-oo,oo) as a periodeic odd(i.e. a(-s) = -a(s) for all s)function with period 2L. Using this function a, we can imbed M as a surface of revolution in E3 if a satisfy the condition [a'(s)I s{ 1. Imbedding di:M -> E3 is given as. . di(p)=(J(iS'i:a;('a')'i'du.,d(s)sine,a(s)6ose) ' ., for p E U where (s,e) is the polar coordinate of p. In the fo11owi'ng, we will use tihe notation p == (s,e>. Thus ip(p) = ￠(s,e). q5(pi) and di(p2) are defined by IeLt'ing s - O+ and s - L.. respectively. q5 is uni'que. up to motions of E3. For the detai1,see [2]. ip is an imbedding of class Ci and is not of class C2 ifthere exists a point po = (so,eo),O < so < L at which a sati'sfies la'(so)i = 1 and a"'(so) l O. Thus if there exist a' such point po stated above, then ip is net of cla$s C2 as long as we adhere that 'di-: M e E3 is a surface of revolution.. So the purpose 'of this note is to' give an isometric immersion th :･ M `> E3 tt which resemble to. ip:M --} E3' and is of class C2. th is given in the next section. '.

(2) Js2. Masao'Maeda. 3 Surfaces of revolution in E3 , In the fbllowing, let M alway's denotes a Si-invariant Riemannian manifold diffeomorphic to a 2dimensional sphere S2 and (s,e) be a polar coordinate such as stated in sectione 2. We also use the same notations stated in section 2. Let ip: M - E3' be an isometric imbeding which is a surface of revelu･tion.. Then ip is given･ as ,, , gb(s,e)=(f(s),h(s)sine,h(s)cQse). for (s, e) E U(O <s< L,O Se< 2T). f and hsatisfies the fbllowimg conditions; , tt t. (a) f'(s)2 + h'(s)2 = 1. (b) h(s) = a(s) (c) f is monotone increasing. ' CQndition ('c) means: that ip i's a:'su'rface of rev61'utiort aro'urid the x--axii an, d (a),(b) means, that di is o,metrc. Frrom (a),,(b),(, e), we ean get precise 'expression of di stated in section 2. '. is'. Now, we consider a mapping ip:M - E3 defined by. ･ th(s,e)=(f(s),h(s)sine,h(s)cose) for (s,e)(,e < s < L,O .<-' e < 2' T)' where f and h satisfie$ the fo11owing conditions;. (a) f'(s)2' +- h'(s')"'.2' --- 1. (b) h(s) = a(s) IFbr cb, we don!t･ reqtiire to ,satisfy j is 'mbnOton,e. i･n.creasing. So imbeddabi'Iity 6f'th does not･ guarantee irr. gen'ar' aL th is an isometri'c. immersion' and, may has interseetion points. But thanks to no requirement of the ･condit'ion that f is monotone increasing, immersion. th : M --i E3 ean get smoothness of class C2' for. certain, Si-invariant manifoId M, .. Note that c(s) = ￠(s,O) is a curve in x -- z plane and cb(M) is a surface given by the rotation Qf. ･the curve c(s) around x-axis. If c('s)' is a graphJof certain function z = h(x), then th(M) i･s a surface･Qf revolution sround x-axis in E3, So, we will call ￠' 'as a revolutionlike immersion. Our assertion is the following,. Theorem LetMbeaSi-invariantRiemannianmanlpldwhichisdwfeomomphictoS2anda=a(s) the function dofned in section 2. ij la'(s)l S1 for all s;OSs <- L and the number of poits s'= so at whieh lat(so.)･1 = 1 is finite, then 'there exists a revolutionlike immersion th. :- M - E3' which･is of class C2.,. Proof Asisstatedbefore,a=a('s)canbeextendedsmoothlyon(-cx),oo)asaperiodicfunction with period 2L. Let {s'i}-oo<i<oo ,; ''' < s･-n-i(= -2L + Sn-1) < s-n(= -L ''i': ptSit) < s-n+1(=. 'Sn-1) <'''< S-1(= '-Si) < So =O < Sl < S2 <'''< Sn-1 < Sn(= L) < Sn+1(= 2L - sn-1> <''' be the sequence of points satisfying la'(si)l = 1 Now,we define f t' [Q,si] -･ R by. f(s)=LS 1--a'(u)2du ' = O. Since a is defined op (--oo,oo), we fbr s E [O,si]. f is smooth on (O,si) and satisfies lims.o+ f'(s). can extendfon [-si,si]. ForsE [-si,so], f(s)' is given as, ' f(s)=-LS i-a'(u)2du . '. This extended f' satisfies lim..o. ,ft(s) = O.. Lemma1 f(s)isdopderentiableats=Oandf'(O)=O.

(3) ON ISOMETRJC.IMIVffiRSIONS OFSi-INM4s,RIANT MANIFOLDS 53. Proof Bythemeanvaluetheoremforintegral, !(,) = (-ii-a"￡･9i,);,g,i fo;,S,O(l.7e.e, ;,9,<,e.`,} .' 'if .) ,O.',. g.hgSh.kiiO.) .--.liMf,?gs)--!.+oStlSitlsf,.(6)limAs-÷o+ i -- a'(eA.s)2 = o･ Aisg we have iL(o) = iim...,- tx.Ei =.. ･･.m. ' is of class Ci on , From L'emma 1, we knew f is of class' Ci on (--si,si).; Successibly, we will show f' Fbr s E (-si,si) r {O}, f'(s) is 'diffbre.ntiable and. fl'(S)=(:1,(isi)S..a)1"i'(i)/S2 ',S<>oO. '. Lemma2 f'(s)･i;･dof)i3rentia.bleatS=Oand. '. tttt ' ' f"(Q)= -a"'(O). Proof RightdifferentialcoeMci'ent'off'ats=Ocanbeealculatedas. ' "(o) == ,.li.m. .. f'(ASIsii, f'(O) '= .},t.ns.., i-Aa,l.(AS)2 (i> Since a'(O) == 1,a"(O) = O because a'(s) attains maximum at s = O. Thus there exists smooth function k near s = O satisfying. . a' (s) =1+ s2k(s) (2). '. k satisfies k(O) = a"'(O)12(-< O). Substituting ℃his in (1), we have. ' .'lttm,'. i'Aa,'(AS)2･='...ii.m,+,i+a'(As) -k(As)lfil,li,l-=/--a"t(o) '. Sim'ilarly,wehave ''. ./. tt. '. f!'"(e)=.}i.m,-' i'-A-a,'(AS)2 =.ltt,-.- i+at(As) -k(A,s)[21f'i = '-at"(o)' Thusf4L'(O)=fL'(Q),== -att,(o).. '. ' -d. '. Lemma3 lim.-o.f"(s)=lim..o.f"(s)=fi'(o) ' Proof Sincea"(s)=.1+s2k(s)by(2), , ' Thus' .fb'S''>O". a"('s) :2sk(s)+s2k'(.s). '. f,.(,)..,' i+-".l' ,s))a' i(s--)zi,(i)=a'(S)i2i-.k,((.S))v)i=E;(gil-Sk'(S)) ((. So. '. .,.ll'{:iF..f"'(s),==VSx/:'wtiii= -a,tt(o).

(4) x. 54. Masao Maeda. Similarly, we have. lim f"(s)= -a'"(O) s-+O-. o , From Lemma 2 and Lemma 3, we see f' is of class Ci and hence f is of class C2 on (-si,si), By the same way, we exPend f on [-si,s2] as. f(,).LSi 1-a'(u)2du-L,S 1-a'(u)2du ' for s C [si,s2]. Then. f'(s)-("ny,!Z,7;gS,)2 ,2i,fs<s2. and f'(si) = O by the same reason as in Lemma 1. We also hEvve f"(s)=(llll/iiliiil￡i(-.i)i, S,i<<,S,<S2. ' ' Since ia'(si,)i = 1, a"'(si) = O.1 and henge. there exists a smooth function k = k(s) 'near s, = s.i satisfying. '. ''. '. '. a'(s)=±1+(s-sD2k(s),k(si)=a'"(sD/2 ' '. So, by the same argument for' so = O,' we have. '. Lemma4 fT(s)isdifferentiableats=siand f"(si)=(: .;,3i'iiS," .,2g(S,il=--',. Lemma5 IiM..,,+f"(s)==lim...,"f"(s)=f"(si> From Lemma 4 and Lemma 5, we see f' is of class Ci' and hence' f is of dlass C2 on (･-si,s2). Successively, we extend f' in the,$.ame w'ay as above as a function f:(---oo,oe) - R. f is given as. ' t/ ''Jti"'1-a'(u)2du---JX2i:a'(u)2du+'";･･+(=-1)i:'"'iX,../1'--a'(u)Sdu,. i' ' f(S)= LJU"-'･ i--a,(u}2du+',c7r,3 i±.,(.)2du+...-F(-i)ji,ft:,,i.-,f[Sf'-.-.i'.S,l'].f),Odr.?OMei'-i. ' ifsE[s,1.r,s,･]forsome2'm<O f ･satisfies the fo11owing:. ' ' Proposition 6 f:(-oo,oo) --> R is s periodic even (i.e, f(s) = f(--s) for all s) function with period 2L and of class C2'. Fbr eabh i; --oo <i < bdJ'(si) =' O and. f"(si)-([:lii .;,{'ilii' .,?gi;zl=-=i,. Proof The periodicity and the evenness of f cQmeS from the periOdicity of a and the evenness of the function 1 -- a'(s)2 respectively. f'(si) and f"(si) are calculated as in the proof of Lemma 1, Lemma 2. and Le.mma 3; ' '' '. Ftom Proposition 6, we have a revolution' like immersion th:M - E3 which is of class C2,. o.

(5) ON ISOMETRIC IMMERSIONS OFSi-INVARIANT MANIFOLDS 55 ' ' tt ' ' Q･EeD･ By relaxing the condition for s surface of revolution, we had a revolutionlike immersion which is of class C2. To make sure of the precise differentiability of this immersion is an int'erestiong problem. Is. thereexistsanintegerr;r23satisfyingthisimmersionisstillofclassCr. -. References . [1] A,Besse, Manofolds All of VWiose Geodesics are Closed,Springer-Ve;lag,Berlin,1978. [2] M.Engman, A IVbte on isometric Embedding of Suofaces of Revotution, Amer. Math. Month,,March. '. [3] M.Maeda and T.Otsuki, Models ef the Riemannian Manofblds 03 in the Lorentzian 4-space, J:Diff.Geom.,9(1974),97-108 [4] K.Shiohama,T.Shioya and M.Tanaka, T7ie GeometTlt of 1lotal Curvature on C6mplete Open Suf:faces,. Cambridge Univ,Press,2003 .･.

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