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(1)Remarks on the Distribution of,Rays tt tttt. By ,, .,: Masao MAEDA* (Received, April, 15, 1981). O. Introduction. Let M bea 2-dimensional complete non-compact Riemannian manifold having non-negative Gaussian curvature K. Then from a well -. known theorem of Cohn-Vossen, the total curvature of M satisfies S.K dv:II2T,. where dv is the volume element of M induced from the Riemannian metric of M, see [4]. Obviously the total curvature of M is not a topological invariant l. when M is non-compact in contrast with compact case. And it seems for the auther that the total curvature of M is expressing a certain curvedness of M.. From this point of view, in [5] we showed a fact as following manner;. For a point qEM, T,(M) denotes the tangent space of M at q. Put Sg(M) := {veT,(M); norm of v=1}. Then form the Euclidean metric on T,(M), Sq(M) becomes a Riemannian submanifold of T,(M) which is the standard unit circle.. Thus we can consider the Riemannian measure on S,(M). Let A(q)cS,(M) be the set defined as. {vES,(M); geodesic r: [O, oo)-> M given by r(t)=exp, tv is a ray}. F. Here exp,: T,(M) -> M is the exponential mapping of Mand geodesic r is called a ray when any subarc of r is a shortest connection between its end points. #A(q) denotes the number of elements of A(q). Under these notation, we have FAcT. Let M be a 2--dimensional comPlete Riemannian manijold wz'th nonnegative Caussian curvature K; diffeomorphic to a Euclidean Plane. Then for. any Point qEM such that #A(q)l2,. measure A(q))-2z-i.Kdv･. t. Note that from non-compactness of M) for any point qEM, it holds #A(q) ;-})1. And for a certain point qEM, called a soul of M, it holds #A(q)}-ii2, see [4]., Note also that from classification by Cohn-Vossen, Mis isometric to a flat open M6bius band or flat cylinder or a one which is diffeomorphic to a Euclidean plane.. The assumption in the above fact that #A(q)l;2 is not necessary, This was proved recently by K. Shiga in [6], And in [6], the above fact was generalized " Department of Mathematics, Faculty of Education, Yokohama National University.

(2) 16 M. MAEDA. for a wider class of non-compact surfaces. Now from above fact and its following remark, it arises a qusestion that. is it possible to give an estimate Of･ measure A(q) from above? And about thisi question, in this note a certain estimate will be given, though it is very. rough and applicable to few manifolds... 1. An estimation. For each value aE(O, 2T], we can easily construct a complete non-compact surface of rotation in 3-dimensional Euclidean space with. non-negative Gaussian curvature K satisfying SMKdv=a and with a point Po GM satisfying measure A(Po)=2T. Thus in the first place it will be reasonable to consider Whether we can give an estimate of l.nMf measure A(q)･. THEoREM A. Let M be a comPlete non-comPact 2-dimensional Riemannian manifold with non-negative Gaussian curvature K Then it holds. ' ,i.n.f MeaSure A(q)-<-3rrrj.Kdv. l. For the proof of Theorem A, we prepare some facts, obtained by J. Cheeger and D. Gromoll in [1], Let P be a fixed point of M. Then from non-compactness of M, there exists a ray r: [O, oo)--M such that r(O)=p. Hereafter, in this note we assume that all geodesics have arclength as their parameters. Then using r, we can constructs a family of compact totally convex subsets {Ct}tio of M which satisfies the following properties;. (1) p=r(O)ECoandift211ti,thenCt,)Ct,and Ct,=}qECt,; d(q, OCt,)l-llt2-ti},. (2) UC,=M, tlO. (3) OCtisatopologicalmanifold(andhenceaCtis homeomorphic to a circle for all t>O). v･. Here a subset CcM will be called totally convex when any geodesic segments connecting any two points of C are contained in C. Fouthermore from Co, we can find a point qoECo called a soul of M which is totally convex as a subset of M. For the detail of these facts, see [1].. In this situation, we have proved in [4] (see Lemma 1) that there exist at least two rays starting from qo. Let a,T: [O, oo)->M are rays starting from. qo and a4T. It sufficies to prove Theorem A when M is diffeomorphic to a Euclidean plane. So, by the broken geodesic T-ioa:(-oo, c>o).M defined by. T-ioo(t)-IZ(,5`' W.h,2".ll.i8,. M is decomposed into two domains Di, D2 such that DiAD2=￠, DdiLID2==M. s.

(3) Remarks on the Distribution of Rays 17 and ODi=aD2= T-'oa((-oo, oo)). For each t>O, OCt is homeorphic to a circle and a(resp. T) meets OCt uniquely at at(resp. Tt). For each t>O and i=1, 2, we set D2:= DiACt. Then in [4] (p. 97-p. 98) we also proved that there exist minimal geodesics a2: [O, m2] - M and bS:[O, n2] - M such that ag(O)=Tt, bt(O)･. at for i=1,2 and al(ml)=bl(nl)EaCt,a?(m?)=b?(n7)EOCt and aS([O,m2]). bl([O, n2])cD2 for i=1, 2. Let Qt be the closed bounded domain with boundary consisting of four geodesics al, bl, b7 and a?. For this family of compact domaines {Qt}tzo, we can choose a monotone inc'reasing subsequence {rt}i..i,2,...C {t}t)o such that. (1) ([?riCQri+i fOr i==1,2,''' and. (2) ,Y.,Qri==M･. l. see Lemma 2 in [4] and its following. LEMMA 1. Illr M is not flat, then for each ri, there exisls ri3.>ri such that eve?y rays starting from any point of the comPlement of Qri,. do not meet Qri･ PRooF. If this were not true, then there exists a monotone increasing sub-sequence {rile}k=i,2." of {rj}d--i,i+i,." and a sequence {qk}k=i,2." such that ri,>ri. and qleE(Qrik)C (==the complement of Qri,) and also a family of rays {rk} le=i,2,.... starting form qk which meet O.,. From property (2), we see d(Q.i, qk). oo as le .oo, where d is the distance function on M induced from the Riemannian metric on M. Let tle be the value such that rk(tfe)EQr, and tk==d(Qri, qk) for le=1, 2, ･･･. Then tk.c>o as le-oo. Since Q.i is compact, we can choose a convergent subsequence {7k,(tk,)}i..i,2,." of {7k(tk)}k=i,2,... Let v be the limit vector. of the sequence {lt,(tle,)}i=i,2,.". And let rco:(-oo, cx)).M be the geodesic given by r.(t)==exptv. Then from construction, r.. isa line through Q.,. Here we call a geodesic c:(-oo, oo)-Ma line when any subarc of c is the shortest connection between its ends points. Then from Toponogov's splitting theorem, M must be isometric to a plane, see Theorem 4.3 in [1]. This is a contradic-. t. tion. q. e, d.. PRooF oF THEoREM A. For any small positive e>O, there exists a number. io such that k. iQr,, K dV)' S. K dv'e.. '. This fact follows from the property (2) for {Q.,}. For this Qri,, we apply Lemma. 1. Then we get Qrj, which satisfies the following; for any point qE(Qrj,)C, any rays starting from q do not meet Qi,. We fix a point qE(Qr,･,)C. If #A(q)==1,. then Theorem A is obvious. So we consider the case #A(q)l2. So S(q)-A(q) is the disjoint union of connected open subsets F2,RGA of S(q), i.e. U F2=. 2En. S(q)-A(q), because A(q) is a closed subset of S(q). For each 2GA, OFR consistst. of two vectors vS vSeA(q). Let r5･:[O, oo)-M be the rays given by r5･(t)=i.

(4) M. MAEDA exp,tvi･, i=1, 2. Since ri and r: are rays, ri and rS do not meet other than q.. Let 6>O be the convexity radius of q. Then from above facts, we get domaines. DR,2EiA whose boundaries are rf([O,oo))Vr:([O,oo)) and which satisfy exp,{tv;vEFR,O<t;S6}CD2 for REA and V D2==M. Then from construction 2EA of (?.i, we can find Dp satisfying DpD(?rt,. Let v, wEOFpt,viEw and a,T:. [O, oo)->M are rays given by a(t) =exp,tv and T(t)==exp,tw. In this situation, we have shown in the proof of Lemma in [5] that there exist minimal geodesics rt,, n,: [O, si]-D,, i==1, 2, ･･･ satisfying the following properties:. (1) rt,,rii:(O,si].D,,i=1,2,･･･, (2) rti-->a,r;i---->T as i--->oo and (3) rt,(O)==r;,(O)==q,rt,(si)==rg,(si),i=1,2,･･･. Since Qri,cD, is a closed set and OD,==o([O, oo))VT([O, oo)), there exists ti6 such that for all tilti6, rt, and rl, do not meet Ori,. Let a(ti) be the compact domains in Dpt surrounded by rt, and n,. Then from above fact, if ti is suffi-. s･. ciently large, we can assume that a(ti)DQri,, Thus applying Gauss-Bonnet thebrem to a(ti), we have. '. S.K dV-e5je.,,K dV $S.-, K dv== ,1/r.m..S,,,,, K dv. == lim {<(72,(O), 7T,(O))+<(-72,(si), -7ii(si))} ti-"'co =<)((6(O), t(O))+ lim <3((-72,(si), -7;,(si))･. ti-co where <(6(O>, t(O)) (resp. <(-7t,(si), -7i,(si))) is the angle between O(O) and e(O) (resp. -n,(si) and -7I,(si)) measured on D... ''. NoW, also from the proof of the Lemma in [5], we see that -72,(si) and. tv tv. -71,(si) belong to the.tangept cone of Ctt at r:,(si)=rT,(si). Here Ct, is one. in the family of totally convex setstv {Ct} used to find {rt,} and {rg,} and satisfying tv OCttDri,(si), Since tangent cone is convex, we have ". <((-7ti(si), -7;,(si));llz (*). For the defip,ition of tangent cone and its properties, see [1]. Thus we hay, e. '.. ', ,',-'' S.Kdv-e:$(S((s(o),t(o))+.;'' .,.･ . '. tt ',..'Ontheot,her,.hg,nq,itholds ,I.' .'',./''.' //" ,' , '... . .' ...<(6(O)f.t(O))=measureF,S2z-measureA(q),. . because F,cS(q)-A(q). Putting this,in the above inequality, finally we have.

(5) Remarks on the Distribution of Rays 19 measure A(q)---e<3z-S.K dv･ Since e>O is arbitrary, we have. '. '. ,i.n.fmeasureA(q)I-{3T--S.Kdv. ''' 'q.e.d. ' '. 4 better estimation REMARK. As is easily seen from the above proof, more for measure A(q) will be obtained if we can give a more better estimation for. <(-72i(si),-7T,(si)),see(*). ･' , , '. /t ' ' 2. A CIassification. Let K be a 2-dimensional complete non-compact. Riemannian manifold with non-negative Gaussian curvature K. Then as is well known, M is diffeomorphic to a cylinder or a open M6bius band when K=-O and diffeomorphic to a Euclidean plane when Ki;O. In this section, we will consider the problem whether some classification theorems of the above type ,. can be obtained by using th'e distribution of rays on M instead of the curvaturg assumption., As a corollary of the Fact mensioned in gO and Shiga's result i'n. [6], we havet.' .' tt ' .. t.. THEoEEM B. Let M be a 2-dimensional comPlete non-comPact Riemannian. manijold with non-negative Gaussian curvatzare K. 111C jMKdv<2T, then the fol-. lowing hold; ', '. (1) There exists a Point p of Mszach that #A(P)==1 ij and only if M is isometric to a flat oPen Mb'bius band.. (2) For any Point P of M #A(p)l2 and there exists a'Point q of M such. that#A(q)==2ofandonlylllCMisisometrictoaflatcylinder. -.-, (3) gOurciza.dneYanPOpZ]natnee Of A4) #A(P)>=3..,I and on.(y z'f.M is dip`igomorphic to a. PRooF. If M.is diffeomorphic to a Euclidean plane, then for any point qEM) v. measureA(q)).2z-S.KdV>Q, 'i"'' ''i Thus #A(q)=oo for any point gEM. So if there` existsapoint pGM such that. l. #MA6(bPiiil tanOrd. #A(P)=2, then M MU9t be isometric .,to, a flat cylmder oqr. eg dqat. REMARK. We can easily construct a manifold sM with SMKdv=2T and ,lth,,P,O.i:lt,S,,gA' sc'."3,:.M,;algS2Y,?,g.,,lli.i("i)ii' #"(P2)=.? a"d #A(p3)-3. Thus ,W. Noticing this, we consider the problem that vs}heh ehe assumption S.Kdv <2rc is removed, are .there exist some char.acterization of a flat M6bius band or a flat cylin' der or a manifold which is diffe6inorphic to a Euclidean plane,.

(6) 20 M. MAEDA. by using the distribution of rays? And a partial answer for this problem is. obtained as follows'. '. THEoREM C. Let M be a 2-dimensional comPlete non-comPact Riemannian manifold with non-negative Gaussian czarvature. Then it hold #A(p)=2 for all Point P of M and inf <)((v., w,)tO where v,, w.EA(P), v,Sw. of and only ij. pEM. M is isometric to a flat cylinder.. PRooF. If #A(p)Eii2 for all point pEM, then Mis isometric to a flat cylinder. or diffeomorphic to a Euclidean plane. We assume that M is diffeomorphic to a Euclidean plane and will derive a contradiction. Let {Qr,}i=i,2,". be the family. of compact domains chosen in gl. Then, just as in Sl, for any small e(T>e>O), there exists a number io such that. S.K dv-E=<:SQ,,K dv For this Qri,, from Lemma 1, we can choose Qrj, such that for any point qE (Qrd,)C, any rays starting from q do not meet Qri,. Since #A(q)=2, S(q)-A(q). =FiVF2. So OFi==OF2={v, w}. Let Di and D2 be the domains determined. `. from Fi and Fb respectively and Di be the domain satisfying DiDQri,. Let. a,T:[O, oo)-M are rays given by a(t)=expqtv,T(t)=expqtw, Then from Lemma in [5], we have. s Kdvl<,(a(o), t(o)). fi1. and S..,Kdv>=<2(6(O), t(O)) where <)(i(6(O), t(O)) (resp. <(2(6(O), t(O))) is the angle between 6(O) and t(O). measured on Di(resp. D2). Thus 2z l.l) S.K dv=S.-, K dv+S.-, K dv. >=<(,(6(O), t(O))+<,(6(O), t(O)) =2z.. So we have. i.K dv==2z, s. S.-,KdV==<i(6(O)', t(O)) and ji, K dv=<,(a(o), t(o)).. Hence. 2T-E=S.Kdv-e==SQ,,Kdv SSs, K dv==<i(6(O), i(O))= 2z-<,(6(o), t(o))..

(7) Remarks on the Distribution of Rays 21 So for arbitrary given small e>O, we have e>`(2(O(O), t(O))=<(Vq, Wq), Vq, WqEA(q), Vq 7! Wq･. I. e. O=inf `b((vq, Wq), Vq, WqEA(q), Vq#Wq･ qEM. This isacontradiction. q. e. d. It seems very interesting for the auther to consider the problem that is Theorem C still true when the assumption inf `5((Vq, Wa) 7E O, Vq, WqEA(q), Vq # Wq. qeM is removed?. References .. [1] J. CHEEGER and D. GRoMoLL: On the structure of complete manifolds of nonnegative curvature, Ann of Math. 96 (1972), 413-443.. [2] S. CoHN-VossEN: KUrzeste Wege und TotalkrUmmung auf Flachen, Comp. Math. [3] B.(iG9R3o5)Mo6L9L-i3a3id w. MEyER: on compiete manifoids of positive curvature, Ann ,. of Math. 90 (1969) 75-90.. [4] M. MAEDA: On the total curvature of non-compact Riemannian manifolds, Kodai Math. Sern. Rep. 26 (1974), 95-99.. [5] M. MAEDA: On the existance of rays, Sci. Rep. Yokohama National Univ. Sec.I 26 (1979) 1-4.. [6] K. SHiGA: On a relation between the total curvature and the measure of rays, preprint in Tsukuba Universi,ty. [7] lilgl-S2Hgi70.HAMA: BUSeMann funCtion and total curvature, Invensions Math. s3 (lg7g),. [8] K. SHioHAMA: A role of total curvature on complete non-cornpact Riemannian 2manifolds, preprint in Tsukuba University.. .. k.

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