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(1)[Science Reports. Some. Yokohama. of the. properties. National. University,. in. geodesics. of. Sec.. the. Riemannian. two.dimensiorral with. Ⅰ.No.. 2, 1953]. large. in. a. manifold. curvature. positive By. Yosio二MtJTO. Intro dtlCtion An. (3 ≦ of. ≦a･). α. class. ry) for. K(P,. and. this point. a. with. Cα will. be. ≧2) of class C糾王′ 9jk (i. k-1, 21･･, ”). tensor. to. satisfy. Hypothesis. P and. for any. two-dimensionalplane. A. if. its. sectional. curvature･. direction. γ through. satisfies. 0<L<K(P,γ)<H. (1) L. where. H. and. are. K. of. e甲reSSion Christoffel. positive. (P･ γ). to. avoid. formed. symbols. K(P,. γ) is given. the. length,. in. lying. the. It is already. 3/4, the. near. dimensional The. in. 1.. Theorem. length≦2l 1) 2) 3) point.. See to. A. or. in. references,. (jib)be. the. tensor. curvature. of. Mn. ,. ,. are. uiv)'ukvl. ,. to. perpendicular. each. other. have. and. unit. a. such. that,. property space. of. is. Mn. manifold. ifエ/〟 is Mn. is. compact(1),. l〇ss than. not. I.. and a. homeomorphic. E.I. number′. to. the. n･. [Rauch]. on. report paper(2). Hopf. Some. theorems. andfuyTLher lhal Mn P, Q u)hick of boinis. A. abair. later･ published loop is a geodesic. ･. in. properties. some. the. concerning. large. Mn. of. are. geodesics. following.. the. Assume has. t:he. Let. as. covering. another. Proof. withotlt. that. valuable. Sn. down. write. γ.. authorwi1l. present. A. (v). universal. sphere. satisfying cited. a. to. (ail)-(,u (ash). (P,.γ)ニーRijkt. direction. known. discovered. Ranch. the R'i･kl. and. de血ition. (”) and. vectors. better. -j款＋(fh). by K. where. gjk. with. (2). be. anymisunderstanding･. ∂ 〈,ih〉. -. then. It: may. constant:s.. Rl;･kl -頂元｢. h. M挿(”. manifold fundamental. positive-de丘nite. said. point. any. Riemannian. complete. ”-dimensional. and. Rinow,. has. a. can. geodesic be. loop(3) of. joined. by. two. wc. (or. Myers･. be. geodesic. arc. whose. initial. point. coincideswith. the血al.

(2) 2. Y.. has. a. closed. geodesic 2･. Theorem. has. Lei. in. curve. eZosed. be. ”. Mn. even. not. 7,/1/￣育∴ lhen. exceed. A･. is orz'eniable. LfMn. lenglh≦2 7,//首ゐco?tyaclible by iu,a. be.joined. can. wh2'ch. Mn. i.. assume. and. wc. w3'ih. Pai'r of Points P, Q. no. i does. of Zengih≦l where lengih≦2 of wc. wcrs. geodesic. more). Mute. io. or. 3Yany. 9oini, Mn. a. ge占desic arcs. length. of. ≦ 7T/I/す･ Theorem pus. afoini makes in MH be. 3･. Assume. ihebole･ a. corresbo71d. and. consider be. Leiサr. set. u'ilh coordi'naies. homeomorphism. a. A. 3.lselfand unit. sphere. P･. I(r). is. arc. from. P. duct,. a. when. is made. of. boundary. Ds(r). r･. C3-1(r) is. From. value. Theorem r. of. ≦27T･//育,. draw. two. Thus. at Jangle Corollary･. we. get. A. Assume. Zengih≦27,//官･. io. ceases. ike. else. touches. (”-1)-dimensional. is. a. of. topological. a. solid. a. joining. arcs. two. (nits. and P. from. homeomorphism loop. pro･. sphere. distance. geod.esic. from. geodesic. homeomorphism. a. a. these. issuing. geodesic. is r.. for. to. P. geodesic. length. arc. with a. arcs. some. point. of art. make. the that. andfuriher. 'Then. a. geodesic and. point. length. arc. is. not. M勿has. 7,//育,. can. of. a. geodesic. ＋, is. the. Cnb-) (ro) u,hick. ∑×Z(r). whose. if. Cs-1 (ro) (with itself) P.. 7T. we. of points. I (r) lo. whichす, or 7,//.-HT. an. the. ifサr. n･disk. that,. >. lhal. ai. (?, 0)･ obtaining. all points. an. know. we. than. for. P'of. ･contact. 3. is locus. the. more. not. for. ro. means. direction. the. r. Mn. ×. touching.. ∑. s means where Mオ･ Although. into. napped. radius. dtEe io such. determines. P. of. bolmdary. singular. necessary.. O≦s≦r. segment. identi'丘cation. disk). be. point. whose. a. is solely. may. explanation. has. ro. g'nio. I(r). ×. io ≡. (a, s). lhe value (into) saiisjies ike inequality. whoseよingulariiy. Some. of. Then. D芸(ro)-中ro(∑ × I(ro)). llmage. ma9blng. ('b,s) with. coordinates ≡. (a, s) belonging. ntcmbers. of. geodesic?oZar. ike. ike. Mn. has. mabbingれis. a. no. loop of. 9eOdesl'c. homeomorbhism. wc. for. r≦. J7r/y/育. 雀1.. Some. theorerrLS. gerLeral. Now. ttlrn. we. tO. the. two･dimensior]alsphere. from. the. just. the. >. an. 4･. 2. Then in. the this. B:. assume. and. following. case. M2. is. homeomorphl･c. will be easily Gaussian curvature. theorems. ordinary. to. a. derived. K. is. Assume. A. and. B.. Then. any. geodes2'c. loop. in. M2. has. arc. 7r//育.. M2. Lei. wiihlength. preceding. be the. m. satisfying. ge?de5ic arc The. for. ”-2. curvatnre.. sectional. Corollary･ in. S2･. precedi叩OneS,. Theorem length. case. A. and. minimum B･ ThenJ. 9oini locus no?oinl. in. u'iih m. can. resbeci. io. bejoined. P. a?oini io. P. by. a. ≦7r/v/官･. theorems. hold. for. any. value. of L/H.. It is. known. that. a.

(3) Some. arc. geodesic. (1),. in. this. have. a. 5.. loop. can. ihe〆xed. of､a. part. vertex. vill. be. the P･. In the. A. Let. the. inner. At. arc near. and. point. each. L/3<. at. x. a. Px.. they. point. from. of. take. two. arcs. ￠1 Rl. x. three. eralwith. Lx. and. beginning these. simple. vben than. between. relation. L/H. in. the Ma. an. Rl. we. two. in. αo. Ql. be. the. just. they reSpeCtively'for Q2 R2 Rl has. infinl'tesimal. Applying. sides. Gauss-Bonnet's. QI Q2 and. RI. R2,. We. tx. get. Px. is. which is. riot. a. is. a. t&,. zero.. to. sonear. above,. Besides,they. have. the. four. rectangular. theorem. smooth point. in. to. a□d. beginning. one. mentioned. re!PeCtively･. 〟. a. singtllar. measured. L∂, not. quadrilateralQI sides,. on. Lx=L′x. tends. x. 0n. length. lies. two･. measured. AlthoughLx is. when. ones. Q2. and. to.. exclude into. was. arc. The. which. Ql, Q2. points. loop. regions･. α(x) with. one. we. 〟2. 7T). point. has. simple. contained. Q∠ R2 R2. and. Its end. a. get. <. 丘Ⅹed. of. locus. this. into. is. at. arc. for. 7T,. (. αo. angle. Its vertical. vertex･. divide. it will. simple. of Lo.. but. coincides. and. to. geodesic. side. consecutive. curved geodesic. be. like) other. number. with Po as be less than. vertical. verticalangle. L3 at. a. and M2 Zo. the. get. the. small,. divides. function. to. loop. Pwill. the. be. the. or. the. loop. being. inner. the. ar白orthogonalto. orthogonal x. Hencethe and. we. geoユesic. any. between. to. consider.a. is taken. which. remains. Lx. lpop. the. we arc. Then. P･. for. angle. point. and. geodesic. draw. ve. double. geodesic. we. If. geodes3'c. at. one. the. (double. loop. which. in. to. continuotlSly. Now. Then. Brst. at. at P,. of ♂. ♂ is said. point. rnoment･エJ. one. obtained. contiuons. αo. ofエo. is. ending. ike. ending. and. a.. angle. simple. by. orthogonally long as px. of. geodesic. a. the. aS. po.. Lo. callthe to.. one,. directed. by. for. we. regions.. for. the. its. geodesic.. lool･s near. tangents. two. shall. we. simple. denoted. geodesics. closed. be. B.. and. be. will. angle. the. signs. not. I/丁≧/￣￣育￣･. among. beginning. part. singular. section. denote. us. can. remarks. a. vertical. no. a. of. verticalangle. can. the. a has. present. satisfyipg. only. length>7'//丁･ 2. any. of. inequality. loop. draw. the丘nal. for. called. curve. closed. loop. We. a.. of. other. tangents. have. near. andfurlher loop. point. conjugate. we. arc. Preliminwy. geodesic. a. the. aT7d. have. geo8esic. simple. consider. the. called. As. 3:. Point ProbZem･. end. us. we. shorlesl. A. Let. has. B. and. be ike. not. Properties. を2.. A. Assume. one. curves. among. Hence. large. length≧7T//￣T. with. loop. geodesic ･. Theorem. arc. points. 27,//￣雷￣. exceeds. gcodesic. end. a. least. at. ⊂Blaschkeコ.. arc. geodesic. its. 301nlng. 2-/-T≧V'首,. ler)gth. in. that. curve. shortest. the. on. points. means. in the. geodesics. has. length≧7T//T. with. of its end. one. of. properties. same. for this. arc. Po･. Let hence are. length vertices. qtladrilat･.

(4) Y.. KdS-JC ∫ is the. JC. where. lies in (&･ We. If. denote. we. RI R2,. of L&. curvature. by. Mut呑. Rl,. at. AA. the. taken. positive. inside. area. its. when. the. concave. quadrilateral. side. and. (I). use. get. LIAA. (3). ･〟<貰岩. RI R2 Now. QI Gland If. Q2)･. we. Q2R2. two. are. RIR2-QI. put. COnSeCutive. geodesics. Q2･り(X),りSatis丘es. Ql. at. parallel. the. (and. differential. wellknown. equation り'/ (X)＋K(”)り(x)-0, is. K(x). where. the. Satis丘es. the. curvature. (4). initial. Q2. COS. (/首x). <RIR2<QI. The. inequalities. forO<x≦7,/(21/育)(fL).. of. L女is. x. smooth. a. O<. closed. In order. Tl(X). in. to. we. get. (3). concave. are. (4). and. that. explain. for small. tx except. toward. (1) andり. satis丘es. (/Tx). Hence. Po. ･quadrilateral. Px. T1(X) the血al. of Lx. part. Px. is. K. positive. the. value. only. has. the. perpendicular Applyirlg respectively･. this. shows. above. was. positive. All such. integralis. singular. that. may. (b) Lx. 7'!(21/首￣), 4). Some ones. of in the. the. be. may. readers. appendix. interior. small. faiI, which_will. fails to. of. a. sides. For. that. su氏cientTy by. two. and. The. x.. rectangular for. isヮ'-α(”). to. point. x≦7,/(2-/￣訂),. the. arcs. beginni昭part. the. Gauss･Bonnet･s. of. decreasing. a. for. considered. consideration. is. (x). α. length. a. such. uniquely. PN Px. at. part. respectively.. at. L∂. arc. theorem. and. we. get. (X)-5KdS the. over. made. thefinal of. geodesic. are. the. on. determined. are. a.-1a. where. both. Its verticalangle. T2(X/)･. T2(x). and. T2(”). Px,. T2(”). T2(X). and PN. Tl(”). Tl(x)and. vertices. Tl (X) and. in. T2(x). point. T2(X). and. at. T1(X). x,. positive. Lo. take. we. of α(x),. a. and. Px. to. perpendicular. ･small. part. Tl(X). arcs. behavior. the. examine. begiming. the. geodesic. related. E(x). Q2COS. x≦7,/(21/m首). curve. As. j㌔.. point. the. for. bounded. and. Rl･. at. conditionり(0)-1,?1'(0)-0,. 0≦QI. positive. M2. of. simple. feel. this paper.. positive occur. loop,. convenient. the. K. quadrilateral･. being. function. value. of. because. wbetber or. (c). to丘nd. ”.. α. If. (a). (x) reaches. proof. of. this. increases. x 〟. exceeds. zero.. inequality. and.

(5) Some. Tbe Lx. occurs. like) for x≦7,/(21/育)･. If (b). decreasing. (b). point. intermediate. an. impossible. (C). of Lx. part. Thus. 0 <. Lx. α(”)<7T.. beginning. occurs. can. not. touches. (a) occurs丘rst. tx,. or. a. else is. former. the. But. latter. the. and. the. mean岳that is. Lx. while. former. the. itself. α(x)-0. Px,. at. that. see. toward. concave. holds. consideration. for. x≦汀/(21/育)･ B. Now. this. we. can. take. Px.. means. line. P(x)Ni. note. Po Ti (X)-yi. to. ＋dx)fro皿a. POint. (”,i)y'i(X)dx,. 2.. or. (x),. T2. Wedraw. the. and. cor･. (X＋d,v), where a. perpendicular de･ Ti (X＋dx)P(x＋dx) and. arc. geodesic. x＋dx･. Ti (X) Ti (X＋dx)where line Ui Vi tO Ti(X＋dx) P(x. a. perpendicular. length. the. means. length. its. and. Ti(X). of. be. will. denoted. by. Ui. VJ. is. Ui･AsTi(x)P(x). Lo, 71t･(X, i) satisfies. to. perpendicular. i. where. 1. means. Ti (X)P(x). On. For. (”)for O<x≦7T/(21/育)･. α. (X)-りl･ (x)y'i (X) dx,. draw. we. Ui. the. Ni. (X), P(x). Moreover. i. following. the. of. a(x). values of x. x and Tl (X), Tl (x＋dx), T2. consecutive. (”)fromP(x). (X)dx.. angle. behavior. (x), P (x＋dx),. In. P(x). -りi. two. P. POints. The. the. examine. we. purpose. respondirlg. y'i. for. the. (c), (x being. toucbes. toward. (b). or. point. or. point.. end. concave. final part. we. and. the. occur丘rst,. the. (a). if. even. double. a. thatエ∬′. see. we. occurs,. that. us. x-x'before. at. of Lx, coincideswith is for O<x<xl. point. because. because. Lx.. until. set. tells. (possiblywith. curve. smooth. a. still remains. (4). (3) and. using. consideration. previous. 5. large. in the. of geodesics. properties. (5)語＋K7&･-0･桝(X･0)-1,昔7E(X･0)-0, for. we. can. use. the. set. the. for. equation. Besides,. deviation.. geodesic. we. getク7'1. (X,X)≡. 桝(”). As get. of equations. cos. just. <. ”. <. rll･(X) <. of the. type. mentioned. previously,. we. (/首i). (X, i) <. (I/了l). cos. also. and. (7). (v/すx). cos. 0 < we. Tl. is. ⊂see appendixコ. (6). for. (5). use. Gauss･Bonnet's. again. P (x＋dx). the part. integral. (x＋dxトα. (x)=α′ (”). is. taken. the. inside. P(x). Nl. over. for. theorem. T2 (X) havir)g. T2 (X＋dx) α. srnall. (/Tx). i≦x≦7,/(21/+育つ･. (X＋dx). where. cos. (X) P(x＋dx). geodesic. P(x)Tl. hexagon. the. sides. ar)d. (X). get. dx-一夕KdS, But. bexagon･. N2(X),. We. as. get. we. can. drop. the highly.

(6) 6. Y.. (8). <. -H(Sl＋S2) Si. where. is. the. dx <-L. tetragon. of the. area. α/. Mut6. (Sl＋S2). Ti (X). As. Ti(X＋dx)Ni(X)P(x).. Si. is. integral. an. 〟. Si-yi/. dx. I 那(X,i). dt. 0. we. from. get. (6). an. inequality. l. Si. 序sin(/すx)y･fdx<. (,,Tx,. <藷sin. y”x･. (8) becomes. and. (9)一昔sin(/Tx,y′<昔<-フ妄sin(/首x,y,･ y. where. (yl＋y2)/2.. means. Comparing. the. in丘nitesimal. P(x) N2(X)P(x＋dx) we. and. we. P(x). triangles. locus. the. observethat. N1. (X) P (x＋dx). of Px. and. bisectstheangle. α. get 1. tan%. -. from. り1yi-r12y3. I. ,. 桝yl(. which, 2. tan子- yi＋772y; ●. ql. Then. (7) gives 1. /了ーx). y'cos(. <. tan. α. 1. <. 2. '. y'cos(/すx). hence. This. inequality. give. and. sin(/￣融) 号<-藷sin(/すx,y,<-藷 cos(/￣￣㌃x)ー￣1. (10) for. y',芸誤覧･ (9) ,A.. cot. C. Now. (ll) and. let. us. take. 6. lie between. α/2,/盲七and -/Ix. An. 7r/2･. for. ike. inequality. and. afunction甲(X). 甲(0)-普. such. that. vcriical. angle. チ, _壁 2. (x),.

(7) Some. sin( /甘x) cot(甲(X) ) cos ( /￣言￣￣x) 1/一房ー｢. pl(x)ニー. then甲is. decreasing. a. long. for. function. Besides,. positive.. as甲remains. From. Hence. can. 9フ(X) But. if. we. not. is. αo. get. vanish. too. from. for. between. x. /首x) cos( /-tx). definition甲(Xo). that. means. we. vanishes,. 1/lj{x). 6.. αo. must. )o. 1/H. inequality. an. satisfy. cos号,喜Jod'2 cos(J%o) A. and B. inequality. loop saiisjies ike if. dO.. o. Assume. example,. dx. cos号-7妄5:ocos(/￣tx) 2. -.og. Thebrem. dx,. get. sin ♂. ･13,. we. Then. vertical. integral. the. as. For. another. to. tends. we. example,. infinite. 2q//育.. as. tends. e. to. 1. rd,′2. cos号>i-5: cos(0/2). An. inequality. already can We. point. P(x). for. obtain. a. for. ike. that. proved. closed geodesic. we considerむst the. de,. sinβ. We. 2;erO,. Then. ,,.. α『ナ7T･. (13) becomes 1. _. de=1wv-A. get. /｢訂'. get. D. was. simbZe. cos. 21/丁-/一宮一･. put. αo. lt. a. cos号,賢Jod'2 (JHie) αn. we. of. (13) becomes. o. -log hence. αo. angle. (13)･. L-H-e,. put. ike. sin ♂. -log. and,. andヮ'/2･. 7,/(21/首■)･ where xo≦7T/(21/育)･. sin(. sin. gcodes3'c For. 0. (12). -log This. and. between. p'>α′/2 (ll) we get. [logp]oXo -7妄一夕言○. by. as. this fact. 7'/2. x-xo,. at. vanishes. small甲(X). <. of甲(0) have. we. remains. 0 and. cos. and,. holds,. if甲≧α/2. of (10), (12) and cot甲≦cot(α/2). for O<x≦7,/(･2H), for α/2 甲(X)>α(”)/2. For. because. O≦x≦7,/(21/育). because. ing. 7. large. the. L. (12). as. in. of geodesics. properties. the. length. which. loop. geod占sic aS. Pl.. The. of. length. arc. inequality. an. simple ”-xl. arc. 1.457.. >. a. se.mote closed geodesic. loop of a geodesic boundsthe length of. and. put. intersections. 7,/(2/育)-xl of Lxwith. the. exceeds a. simple. ,. denot-. geodesic.

(8) 8. Y.. T1(X1) Pl. arcs. meaning. T2(Xl)Pl. and. arc. Lx. which. denoted. by. L半(x), whose. the. respectively, are. previously,. asgiven. remains. Mtlt6. M1(X). named. Tl. where. M2(”).. and. P(ガ) Mi(X) erasing by Pe(x). Then the. after lerlgth. T2. and. and. have. the the. while. M2(X). P(x). integral. same. part. of. will. be. curvature. of. L#(x). along. iNゐ-◎(x). (14) is related. l#(”) by. to. me(x). (15). ---@(x).. dx Consider. Ml(X). an. L弗(x＋dx). arc. M1(X),. M1(X＋dx) are. vertices. L尊(x) and. near. M1(X＋dx). Mi(”) M2(X＋dx). and. and. rectangular,. curved. a. geodesic. withshort. long. and. make. sides, L滞(x) and Gauss･Bonnet's theorem. ctlrged. applying. quadrilateral. M2(a) M2(X＋dx) L雅(x＋dx). Its. sides,. we. get. ◎(xト◎(x＋dx)＋dxJ戯お-0 ,. int:egral. the. where. is taken. (16) and. (16). the. along. Ll#(x) (15). <慧<. arc. L#(”).. Because. of. (1). we. get. HL*(x)･. give. %<-Ll* (17). %> the. while. initial condition. for x-0,. for, Lo. qtlalities. (17) 'together. being. -Hl# is. a. geodesic,. <. this,. we. Z弟(x)<. 7妄lo･. we. apply. show. Esee. differentialineappendix. lo*. cos(/Tx). <雷,. get. sin(/首x). Now. The. @(0)-0.. (16)gives. L15 cos(/首x) integrating. know. we. initialcondition. withthe. Then. forx≦7T/(21/育)･. l#-lo.･. T告-o･. lo*cos(y'7{x). and,. K. Gauss･Bonnet'stheor占m. <. @(”,. ･. far L(xl) and. obtain. 2j.

(9) Some. properties. in the. of geodesics. large. 9. 5KdS＋@(”,)＋7'-α(Xl)-27'. Then. because. 珍(xl)<27T. get. we. of. lob< 2打･. 7LfThis. loops, for simple geodesic proved follow･ an almost similar and. was. But. we. can. simple. and get the geodesic A and B. 7. Assume. closed. Theorem in. geodesic. M2. sail.Sjieslhe. hence. α<7r,K>O,. for. not. rather. simple. Then. ike. Thus length. arc. geodesics.. treatment. simpler. inequality.. same. closed. we. lo. a. a.. the. get. of. for. slmble. closed. for. argu･. inequality H. lo<窟丁･ Appendix First:. the. compare. we. 1 differential. of the. solutions. equations. an. s,. ment. E//＋HE-0,. り//＋K?1-0, the. with. initial. conditions. り(0)-E(0)-0,?1′(0)-E/(0)-1 where. E(s). stant,. we. (1),. satisfies. that. 0<. is,. ,. H. L<K(s)<H･. being. a. con-. positive. have l. Eニー序sin(/首s) We. is not. If this. り(So)-ど(sJ)>O su氏ciently α. denoted. We. than. α1.. α1.. equations. αり> E. hence. st:atement:. We. α1. for. Tl(s) >E(s) But. s.. positive. a. O<s<. then. for. we. so, can. O<s≦so,. of linear. are. for. some. value. type,. of. we s. near. for. we. have. a. small. positive. greatest. of. the. for some. get s1.. but. O<s≦so,. value we. of. compare. the. solutions. of the. wotlld. り′′＋K,7-0,. have,. αlTl′/--Kalケ7 Contradiction. E′/＋LE-0,. constant. as. equations. As >. is less. α1. 0. an. α. is. α. such. between. s. αり(S)≦E(s).. differential. for. surelyり>E. is true. next. for which. 7T//育,. 丘nd and. O<α1り(s)≦E(s). Satis丘es. 0<so<. so,. number. for α1ク7(so)<E(so). Hence have must α1り(s')-E(sl),for, otherwise we from Moreover get α1?7′(Sl)-E/(sl). differential s=sl,. 7T//￣育∴ have. must. have. we. s-sl,. while. we. O<αり(s)≦ど(s). by 1. true. small. that. such. than. for O<s<. tha=7>E. prove. I. andso, greater. the starting. -HE-E//. occurs. and. at. the.

(10) 10. Y.. initial conditions. the. with. Mute. り(0)-E(0)-0,り/(0)-E/(0)-1, K(s). where. OくL<K(s)<H.. satis丘es. We. l. E -7〒-sin( we. and remains. Of. positive･ is not. If this. we. We. for. /Ts) interval0<s<s6<7,//￣Tfor. any. have. we. 0 < a. consider. a. set. have. we. ofal1. For. O<s≦so･. for. the. り(S)≧αf(s) for some number 17(sl)=αE(sl),り′(Sl)-αE′(sl) henceり:s) < αE(s)for get?I′′(Sl)< αE′′(sl), tradicts. the. Now. starting. we. assumption the. compare. the. and. set,. which we. get. 0 and. so.. Then. we. near. sl.. This. con-. is true.. statement. of the. sohltions. interval. some. this. of. between. sl. (α) for. numbers. number. whichり(so). for su凪ciently. a. rl <. surely. positive. greatest. for. 0<so≦s6,. so,. number. <so.. s. whichり. s6> 7r/】/-HIR.. have. must. a(s) for. <. s･. positive. course. true. j=E(so),whileり(s) small. for. thatりくE. prove. have. differential. equations. E′/＋HE-0,. り′/＋&7-0, ●. with. initial conditions. the. り(0)-E(0)-1, We. that. verify. thatり>. If this. is not. for. true. O<s<7,/(21/首), have. we. s･. small positive E for O≦s≦so･. which叫< we. If. arrive we. at. a. again. for. ”(so)-E(s8),Whileり>E. su琉ciently. and. E. 77'(0)-E/(0)-0.. We. 0 <. so,. number. O<s<so,. for a. consider. Further. E-cos(/首s).. while. set. have. for surelyり>E positive numbers (α) for as will follow just above. we. of. consideration. such. so≦ 7r/(2v/￣訂),. contradiction.. the. compare. solutions. of. a//＋LE-0. り'l＋K7-0, for. り(0)-a(0)-1 ,り′(0)-a/(0)-0 O<s<7,/(′2-/宮う, while卓-cos(/Is). ,. we. getり<f by. obtained We cause. can. the. of. for. the. cbanglng extend. absence. foregoing. interval. the. of equality. a. proof. 0 <. Let. us. compare. a. to 0 < < 7,/(21/一育一) in (1). signs. functionク7(s). is. proof. little.. s. Appendix. Its. s≦7,/(21/育). be･. 2. of. in equality. 77!/ > -Hq. class. C2. satisfying. a. differential.

(11) Some. is the. solntion. If. in. large. the. ll. which. of. E/′ニーHE, we. geodesics. initialconditionり(0)-0,り′(0)-1,with E(s)-sin (/-万一s)//育,. the. and. of. properties. E/(0)-1.. E(0)-0,. put. v(s)-り(S)-E(s) get. we. ktlOW. we. and. positive. =o,. so. number. ”. has. <. sc,. ≦s2. that. the丘rst. vl[(s) > -Hv,. ”(0)-”/(0)-0,. ” >O. for. small. such. that. have. we. maximum. at. s-sl. vu>-Hv> for. O≦s=<sl,. hence. We. Zero,. the. and. v(sl)-ml. s.. Hence. there. so･. If. have. we. last maximum. at. and. Then. ”(s2)-m2･. -Hml. integration. by. v/(sl)-Vl(s). ”/(sl)beillg. positive. > 0 for O<s<. ”. Put. coincide･. may. Which. sufBciently. >. (sl-S).. -Hml. get. Hm1. v[(s) <. (Sl-S). ,. hence. for. v(s) , have. we. sl. Almost. ÷Hs21]･ -l[1-. the. same. >J%･. we. gives. consideration. so-s2. >. ー≦｢. ノ首'. get. >. so. proves. the. validity り>. for. [1-÷H(sl-S,2]･. is,. that. This. -i. hence. v(0)-0, o ,. and. (sl-S)2. Hml. Thisgives. O≦s≦sl.. while. ÷. <. ml-”(S). O<s≦7T/(2/育)･. 21/￣官￣/I/￣育￣･ of sin. (/￣訂s)//育. s-s2, we. is. a. v(so) 0<sI get.

(12) 12. y.. A. argument. similar. that. shows. Mute. 77(s) of class. C2. satisfying. ?1′/<-Lり,り(0)-O,り′(0)-1 tsatis丘es sin. り< for. O<s≦2/1｢//了￣, Similarly. we. hence. for. O<s≦7,/(2-/-訂),. if. we. have. H>L.. get cos. り> ずor 0 <. (/￣￣王￣s)/I/T. (/TFs). s≦7T/(21/育￣) if-でis. a. solution. of. り′/>-H77,水0)-1,り′(0)-0, and cos. り< for. if?7. O<s≦7T/(2/育). (/￣盲￣s) is. a. solution. of. り//<-Lり,り(0)-1,り′(0)-0, H>L.. and. References W. a Y.. B)aschke and Fほchen,. Mut6:. S･ B･ 1L. to. Myers:. E･ Rauch:. (1951), J･ L･ Synge:. Vorlesungen也ber. :. Hopf. W.. Comment.. Math.. be. published RieHlamian A. contribution 38-55. On. the. (1936),316･-320.. Differentialgeometrie. iJber den. RinodT:. Helv.,. Begriff. 3. der. I, 1930. voIIst払digen. differentialgeometrischen. (1931),209-225.. later. manifolds to. connectivity. in the. diHerential. of spaces. large,. Duke. geometry. of. positive. Journ･ 1 (1935), 39-49.. Math･ in. the. large,Annals. curvature,. Quart.. of. Journ.. Math.,. 54. M■atll. 7.

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