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(1)A Note on the Geodesic Circles and Total Curvature By. Masao MAEDA* O. Introduction.. Let M be a complete non-corhpact Riemannian manifold. For a point pEM and non-negative t;-})O, let Bt(P):= {qEMId(P, q)$t} be the geodesic ball with radius t centerd at P and St(P):==OBt(P) the geodesic sphere (circle when dimension of M=2) with radius t centerd at P. Here d is the distance function in-. duced from the Riemannian metric of M. And put. D,(p)2 Dt(p):=ij/iililm. t ' Then, in [7], we have showed that the number lo defined by lo:= i'i/ifim Dt(P)2. t'co t. '. does not depend on the choice of the point P which was used to define it. Thus. lo is a number determined by the Riemannian structure of M. And a meaning of lo is also shown in [7]. By definition, lo can be infinity. And the purpose of this note is to show that for a certain manifold M the condition for the. '. constant lo such that. lo< oo imposed a strong restriction on the Riemannian structure of M. It is stated as follows;. THEoREM. Let M be a comPlete Riemannian manijold dofeomorPhic to an Euclidean Pgane E2 and with non-negative Gaussian curvature K. If the constant l, of M is finite, then total curvature S.Kdv of M equals to 2rc.. Here dv is the volume element of M induced from the Riemannian metric of M.. 1. Proof of the Theorem. For the proof of the above theorem, we need some preparations. LEMMA 1. 71here exists a Point PEM for which following case (A) or case * Department of Mathematics, Faculty of Education, Yokohama National University..

(2) M. MAEDA. 2. (B) occur;. Case (A). ･ .. there exist two rays ai: [O, cx))-M, i=1, 2, starting from P such that. -a,(o)==6,(o) (in this case M is divided into mutually disjoint domaines Di and. D, such that DiVD2==M and ODi=OD2=6i([O, oo))Vo2([O, oo))･ Case (B):. there exist three rays ai: [O, oo)-->M, i==1, 2, 3, starting from P satiEuCying the following; by oi, i=1, 2, 3, M is divid2d into mutu-. ally disy'oint domains Di, D2 and D3 such that DiVD2VD3 =M and ODi= ai([O, oo))Vai+i([O, oo)), i= 1, 2, 3 (in the following, we con-. sider the numberi nzod 3. Thus a4= ai). Then. <}(D,(6i(O),6i.,(O))<rr fori=1,2,3, Here a geodesicc: [O, oo).-M is a ray if any subarc of c is a shortest connection between its end pomts. ' Inthisnote,letallgeodesicshavearc-lengthastheir. parameter. And <Di(v, w) denotes the angle between the vectors v and w measured on the domain Di.. PRooF. This was essentially proved in [4; Lemma 1, pp 96tv ]. There, we have proved that there exists a point PEM called a soul such that for any non-zdro tangent vector vETp(M) (==tangent space of M at P), there exists a ray a:[O, oo)->M starting from p and satisfying. 'n7, <(v, 6(O))S ' where <(v, w) is the angle between the vectorsvand w. Since Mis noncompact, there exists a ray ai: [O, oo).M starting from P. Thus, putting v:= T6i(O), we can .find a second ray o2: [O, oo)-->M starting from P and <(-6i(O), 62(O));$T/2, i.e. <((ai(O), 02(O))}IT/2. If -6i(O)==62(O), then case (A) occurs. So, we consider the case -ai(O)#a2(O). By ai and a2, M is divided into two mutually disjoint domaines Di and DE such that DiVDE=M and ODi==ODE= ai([O, oo)')Va2([O,. oo)). Since -Oi(O)7e02(O), we can assume, renumbering if x. necessary, 'li- s <D,(o,(o), a,(o)).. Again, by putting v:= -(6i(O)+62(O)), we get a ray a3:[O, oo)--->M starting from P such that <.E(v, 63(O)) ;;lz2,. Then ,. by putting 0o:=`)(D,(Oi(O), 62(O)), we have.

(3) A Note on the Geodesic Circles and Total Curvature 3 0i:==<D,(6i(O), b2(O))+<l(.s(6'2(O), a,(O)). l.}ir 0e +. z-0o 2. == g' zo' If ei<x, then applying the same argument for the vector v:=-(6i(O)+63(O)), we get a ray a4:[O, oo)->M starting from P such that <DS(V, 04(O));iE{ -li- .. Then e2:== 0i+ <)(DL(b3(O), 6,(O)). rt-ei T 0i. )0i+ 2 ==i+i e, zz 5+2-,+2-,･･ If 02<z, then again we can find a ray as: [O, ep).M starting from p such that. <.E(V, 6s(O)) :Ill -2- ,. tt. where v:==-(6i(O)+64(O)) and e3:= e2+<[[1.s((r,(O), 6,(O)). z-e2 n. 02. lO2+ 2 ==i+i ig+i,-Z, +,-Z, +tyo,･. Continuing this method, we can consider the following two cases; Case (I): there exists a sequence of rays {oi}i=i,2,3,." starting from P such. that -. <DE(-(ai(O)+6i+i(O)),6i+,(O))211{i- .' . for i=1, 2, 3, ･･･ and putting. t tt. 0i:=<D,(6i(O), 6,(O))+<D6(6,(O), 6,(O)) + ''' +<DII(ai+i(O), 6i+2(O)),. 0o<0i< ''' <0t-i<0t< ･'' <T. and oi l-ir -li- + 2-Z,2 + '"'+ tt + tYl!･. t .t.

(4) 4 M. MAEDA. Case (II): using the above notation, there exists a nember io such that. 0o<0i< ''' <0i,-i<Z:;llOi,･ If case (I) occur, then the limit geodesic a:[O, oo)-M of {ai}i=i,2,.. is a ray starting from P such that. O(O)=='Oi(O), because 0i->T as i--oo. So, renumbering a2:=a, this case reduces to case (A). In case (II), if T=0i,, then renumbering a2:=ai,+2, this case also reduces to case (A) and if n<0i,, then renumbering a3:=ai,+2, this case reduces to. case (B). q. e, d.. Now, let assume that case (B) occur and ai: [O, oo)->M, i=1, 2, 3 are the rays starting from P having the properties mensioned in Lemma 1. When case (A) occurs, we consider two rays oi and o2 having the properties mensioned in Lemma 1. And as we note in the following, our proof in case (B) goes well in the same way in case (A) by restricting i=1, 2. For each tlllO, put. Ct:= A (M- V B,(c(t+s))),. e s)O. where intersection is taken for all rays starting from P. Then Ct is compact totally convex subset of M with boundary 6Ctwhich is a submanifold (possibly topological) of M. Here a subset CcM is called totally convex if for any points q, q'GC and for any geodesic r:[O, d(q, q')].M from q to q', r([O, d(q, q')])cC.. These facts were proved in [1]. Since dimension of M= 2, each OCt is homeomorphic to a circle. And in [4], it has proved that for each t>=O, each ray starting from P meets with OCt only at one point. Thus, for each t>O, each ray ai, i=1, 2, 3 meets with 0Ct only one point ai(t). For each t>O, put. D3 := D, A C, and. Ee:= D,AOC, for i=1, 2, 3 (i=1, 2 in case (A)). Then, by the above observation, each DZ is a domain whose boundary ODt=ai([O, t])VESVai+i([O, t]) is a Jordan curve in M for i=:1, 2, 3 (i==1, 2 in case' (A)) and t>O.. LEMMA 2. For any Point qE!De, any shortest geodesic a:[O, d(ai(t), q)]->M from ai(t) to 4 does nol meet with ai excePt ai(t)==a(O) for i-'-1, 2, 3 (i=1, 2 in. case (A)) and t>O.. PRooF. If a meet with 6i at a(to)==ai(t-to), O<to<4 then, since ai is a ray and a is a shortest geodesic, we see that a [O, t] and ai [O, t] must coin-. side, i.e. a(s)=ai(t-s) for O;Ss;St. Hence dv(t)==-ai(O). By assumption, <((D,(6i(O), 62(O))<T. Thus, from these facts, we see that for sufficiently small. e>O, a((t, t+e))cD?. Since OD7 is a Jordan curve and q is an exterior point.

(5) A Note on the Geodesic Circles and Total Curvature 5 of D?, a must intersect OD2. So, since Ct is totally convex, a must intersect a2((O, t] at a(t+ti)=a2(ti), O<ti<t. Thus, there exist two shortest geodesics al[t,t+ti] and ol[O,ti] from o2(O) to a2(ti). But this is a contradiction, because a2 is a ray, When case (A) occurs, dv(t)==-6i(O)=a2(O) and hence a)[t, 2t]. =a2. Soadoes not through the point q. q.e,d. From this Lemma 2, we can prove the following Proposition.. PRoposlTIoN 3. For each DS, i=1, 2,3 (i=1,2 in case (A)) and t>O, it holds that for each Point qEDt, there exist shortest geodesics. ai: [O, d(ai(t), q)] --> M and. bi: [O, d(ai+i(t), q)] . M from ai(t) to q and ai+i(t) to q resPectively such that. ai((O, d(ai(t), q)))cD2 and bi((O, d(ai+i(t), q)))cD2.. PRooF. For the simplicity, we will show this when i=1. 0ther cases are proved completely similar. Let consider the following subset F of D} defined by F..{qED} ;?g.re 2i,ls,t?,a,sgo.zt,esl,g,eO.d,?S,iC,f.:,g,9; ,d,la,e('is,q']""}･. We will show that F is an open and closed subset of D} with respect to the relative topology. Let {qle}le=i,2,...CF be a sequence such that qk-->qoED} as le->oo. Then, there exists a sequence of geodesics {ale}le=i,2,... such that crk:[O, d(ai(t), qle)]-M is a shortest geodesic from ai(t) to qle and crk((O, d(ai(t),. qk)))cD} for le=1,2,3,･･･, by definition. Then, choosing a subsequence if necessary, we can assume that ale-ao as le.oo where cro is a shortest geodesic from ai(t) to qo. Since ak((O, d(6i(t), qle)))cD2 for le=1, 2, 3, ･･･,. ao((O, d(ai(t), qo)))cD}.. And hence, by Lemma 2, we see that ao((O, d(ai(t), qo)))cDl.. Thus qoEiF. Hence F is a closed subset. Next, we consider the set 17C. By definition, qEFC--if and on!y if any shortest geodesic a:[O, d(ai(t),q)]->M from ai(t) to q, a((O, d(ai(t),q)))aD}.. Andalso,bytheproofofLemma2,weseethat ' ' a((O, t'))cDZVD?Va3((O, t)) (in case (A), cr((O, t'))cDZ) for some t', O<t'<d(ai(t), q･)). Now, let'{qle}le..i,2;.i. cFabeasequencesuchthat '･' '' --'.

(6) M. MAEDA. 6. qle.qo asle-->oo. Let ale:[O, d(ai(t), qfe)]--->M be a shortest geodesic from ai(t) to qle, le=1, 2, 3, ･･･.. Then, for each le=1, 2, 3, ･･･, there exists a number tle, O<tk<d(ai(t), qle) such. that ak((O,tle))cD?VD7Va3((O, t)) (in case (A), ale((O, tk))cD2), By choosing a subsequence, if necessary, let. ak-ao asle-oo. ao:[O, d(ai(t),qo)]--÷M is a shortest geodesic from ai(t) t6 qo. So, again by. Lemma 2 we can see that ' ao((O, to))cD2vD?Va3((O, t)) (in case (A), ao((O, to))CD?) for some to, O<to<d(ai(t), qo). Thus qoEFC and FC is a closed subset, i.e. F is a open subset. So F is open and closed subset of. D}. If qeDl is sufficiently near to ai(t), then clearly qeF. Thus Fis nonempty. Hence by connectivity of D}, we have. D}=F. Again, by the same argument, we also have D}..{qGD} IPs.re sx,,is,t2,a,sgg,rkeit,,g,eo.d,?slc,f6:,,EO,; ,d,ga,&(t)",q)]-'M}. Thus Proposition 3'is proved. Q, E. D. Proof of the Theorem. Let PEM be a point having properties mensioned in Lemma 1.. Since. Io= i'i/ifim Dt(P)2 <... t.oo t we see li.m. Dtt(P) =o. For, if there exist a sequence {ti}, t" oo as that 1,irim.D`. i->oo sUch. ;:.P)>o, then, Dt,(p)-->oo as i->cx), Thus ''' p/i}:i[m. DtlP)2 )F. ii:iii.m. DtE.P)2 =li.m. Dti. P) ･D,,(p)=oo･. This is a contradiction, so lim Dt(P) =o.. t'co t Now, by the definition of Ct,. S,(P) (=OB,(P))c C, for all tlO. Then, applying Proposition 3, we see that for all t>O and for all point qGD2ACt, there exist shortest geodesics.

(7) A Note on the Geodesic Circles and Total Curvature 7 ai: [O, d(ai(t), q)] . M from ai(t) to q and. bi: [O, d(ai+i(t), q)] - M from ai+i(t) to q satisfying. ai((O, d(ai(t), q)))cD2. and bi((O, d(at+i(t), q)))cDg. for each i=1, 2,3 (i==1,2 in case (A)). For each･i and t>O, fix a point qiE DeACt and let Ti:[O, t]-->M be a shortest geodestic from P to qi. Then, for each i, Ti((O, t]cD2. This is easily seen by a topological observation and the fact that Ct is totally convex. For each i and t>O, let Ai:==Ai(ail[O, t], Ti, ai). and Al･ :=A{･(ai+ii[O, t], Ti, bi). '. are the geodesic triangle with three sides ai][O, t], Ti, ai and ai+il[O, t], z'i, bi. respectively. Then, for each iandt>O, D2==AtVA;･ and AiAA:･ =Ti([O, t]) where Ai and Al･ are considered to denote the compact domaines bounded by Ai and A{･. respectively. Now, applying Toponogov's Comparison Theorem (see [3;pp 183rv ]) for these geodesic triangle Ai and A{･, we have two triangles Ai= Ai(A, B, C) and AC･=A:･(A', B', C') in an flat Euclidean plane E2 such that. (i) AB==AC=t B C = d(o t(t), qi). (21gl/S,'8,li,1, ,,)). and (ii) IA:El<D,(6i(O),ti(O)) ,LB$<(.,(-6,(O), a,(O)) ,,L c $ <)([.,(t,(o), a,(o)). ,C At:S <.,(6,.,(O), ti(O)) zC Bi:S<[(.,(- 6,.,(O), bi(O)) zI C';$; <.,(t,(O), b,(O)). respectively. Then, by definition, we have I B = zl C =cos-'(d(ai(t), qi)/(2t)). lcos-i(diameter of St(P)/(2t)). - cos-i( DslP)).

(8) :. l l. 8 M. MAEDA and also. :. z B'= z c';i; cos-i( DSIP))･ Thus, from previous observation, we haveb. IB=ztC.{} qst->oo and. ' IB'==zC'.g ast-oo. By applying the Gauss-Bonnet's Theorem to Ai and AC･, we have S,,Kdv=<D,(6i(O), ti(O))+<.,(-6,(O), d,(o))+<.,(o,(o), a,(o))-z 2-i <g(D,(6i(O), ti(O))+ z B + ,ttC -z. and also S.1.KdV= <)(D,(ai+,(O), t,(O))+ ,t B,+ .< ct-.. for each i and t>O. Thusi, for each i and t>O,. tt. Ssl,Kdv=S.,Kdv+S.tKdv. }ll <)([D,(Oi(o), ai.,(O))+ Z B + Zl Bt+ ,<t C+ Z Ct-2T.. So, since ZB, ,LB', zCC and ,LC'.rr/2 as t->oo, we have ji,I. K dv = 1,Lm.. S4, K dv. lll;<[(D,(ai(O), 0i.,(O)). for each i. Thus, we have. S.Kdv= >i.]Sbi.Kdv >- Z<(D,(bi(O), O,,,(O)). 1 =2z.. On the other hand, by a well known result by Cohn-Vossen (see [2]), it holds. s Kdvg2T. M'. Thus, we have proved that. j.Kdv=2z. Q. E. D..

(9) A Note on the Geodesic Circles and Total Curvature. 9. References [1]. J. CHEEGER and D. GRoMoLL: On the structure of complete manifolds of non-nega-. [2]. S. CoHN-VossEN: KUrzeste Wege und Total KrUmmung auf Flachen, Compositio. tive curvature, Ann. of Math., 96 (1972), 413-443. Math., 2 (1935), 63-133.. [3]. D. GRoMALL, W. KLiNGENBERG and W. MEyER: Riemannsche Geometrie im Grossen, Springer-Verlag, 1968.. [4]. M. MAEDA: On the total curvature of non-compact Riemannian manifolds, Kodai Math., Sem., Rep., 26 (1974), 95-99.. [5] [6] [7]. : A note on the set of points which are poles, Sci., Rep., Yokohama National University, Sec. I, 32 (1985), 1-5. : Geodesic circles and total curvature, preprint.. : Geodesic spheres and poles, preprint..

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