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(1)A Note on the Set of Points which are Poles by. Masao MAEDA* ' Let M be a complete connected Riemannian manifold and expq: Tq(M)-M the exponential mapping of M at point qeZll where T,(M) denotes the tangent. space of Mat qcMl Point pEMis called a pole when expp: Tp(M)-M is of maximal rank at every point of Tp(M). And as is well known, existance of a pole gives a strong restriction on the topology of M] In fact, if PGM is a pole,. then expp: Tp(M)---->M is a covering mapping and hence the universal covering fi of M is diffeomorphic to an n-dimensional Euclidean space T,(M)xR", where. n=dim M: This is so called "Hadamard-Cartan's Theorem", see [4]. From the definition' the condition that point pGM is a pole is equivalent the condition that for. any geodesic c: [O, oo)-->M starting from p, there exists no conjugate point on c to P along c i.e. there exists no Jacobi field X:O along c satisfying X<O)=O and. X(to) =O for some toe(O, oo). Thus, by taking the second derivative of the square norm of a Jacobi field along a geodesic, we easily see that. all points of non-posi-. tively curved manifolds are poles. Thus universal covering manifolds of non-posi-. tively curved manifolds are dieffeomorphic to some Euclidean spaces. And from this reason, many properties of non-positively curved manifolds are drew out. But. on the contrary to negatively curved cases, most examples of positively curved. manifolds have only few poles. Of course, because of M.Morse-I.Schoenberg's Theorem, see [4; pp 176] (with respect to Ricci curvature, this result is known. as S.B.Myer's Theorem, see [4; pp 213]) positively curved manifolds can have poles only when they are non-compact. And through these examples and geomet: rical meaning of the curvature, the auther considerd the following problem: Is there exist a positively curved manifold having many poles?. The purpose of this note is to give a positive answer to this question.. Let M be a complete connected Riemannian manifold of dimension n and P the set of all points which are poles., At fir$t from the definition of pole, we. easily have the following. ' Proposition 1. P is a closed subset of M. Proof Let n: tN, M-Mbetheuniversalcovering of Ml Let c: [O, oo)-M. " Dept. of Mathematics..

(2) 2 M. MAEDA. be a geodesic starting from a point pGM] a: [O, oo)-M Na lift of c starting from a point Pez"i(p) (Throughout this paper, we assume that parameters of all geodesics are arclength) and X a Jacobi field along c satisfying X(O)==O and X/(O). =O, where "'" denotes the covariant derivative of X along c. Then for the Jacobi field X along cN satifying X(O)=O and clz(X/(O))=X'(O), we have dT(X)= X, where dz is the differential of z. From this fact, we easily see that. Ptv= z-'(P). .(. tv tv tv tv. where P is the set of points which are poles in M. So it suthcies to show that P is a closed subset of M because n is a local homeomorphism.. tV tV N is By Hadamard-Cartan's Theorem, for any point PeP, eKpp'-: Tp"-(M)-M a diffeomorphism. That is, any geodesic cN:[O, oo).A2i starting from P is a ray. Here by definition cN is a ray when any subarc of cN is a shortest connection. tv between its end points. Let {Pi}i..i,2,...cP be a convergent sequence such that. tv tv. Pi.PoGM as i-cx). For any geodesic cN:[O, c>o)-M starting from Po, we can choose a sequence of geodesic cNi: [O, oo].A2i such that cNi(O)=Pi,i--1,2,･･･. which satisfy cNi.i So, since each cNi is a ray, a is a ray. This shows that Po. isapole. q.e.d. A geodesic c:(-oo, oo)-M is said a line if any subarc of c is a shortest. connection between its end points.. Now we have the following. '. Proposition 2. If Mis simply connected and have no lines, then P is a compact. subset ofM ･. '. Proof Assume that P is not compact. Then, from Proposition 1, P is not a. bounded set. Thus there exists a divergent sequence {Pi}i=i,2,...cP i.e･ d(Pi,Pi). -oo as i-c>o. Let ci: [O, d(Pi, Pi)]-M be a shortest geodesic connecting from Pi to Pi, i=1,2,･･･. Since pi=ci(O) is a pole, from Hadamard-Cartan's Theorem, each geodesic starting from pi is a ray. Thus extended geodesic ci: [O, oo) x'. -M of ci is a ray passing through pi= ci(d(Pi, Pi)), i=1,2,･･･. From the sequence of unit tangent vectors {ei(d(pi, pi))}, i=1,2,･･･ at Pi, we can choose a convergent subsequence {ei,.(d(pi, pid)}y=i,2,･･･ and its limit vector veTp,(M),. eij(d(Pi, Pij))-->v as 7'-oo. Then the geodesic c:(-oo, oo)-M defined by. c(t):=expp,(tv) isaline, by construction. q.e.,d. Without the assumption that M is simply connected the above Proposition does not hold. For example, fiat cylinder is such one. But furthermore if we '. giveacurvaturerestrictiononMthenwehave; , .. Theorem A, Let M be a complete connected Riemannian manifold with non-. ]i. aj i,. '1. e.

(3) ANote on the'Set of Points which are Poles 3 negative Ricci curvature which is positive at some point of Ml Then P is a. compact subset of M: ･ Remarle. Ricci curvature is, by definition, positive at point peM if for any vector veTp(M), v )FO, Ricci curvature in the derection v is positive.. Doof: Let T: fi---->M be the universal covering of lttZ Then for any geodesic cN i'nNM, TocN is also a geodesic in M] because n is a local isometry. Thus, if. -. L. d and d is the distance functions on M and M respectively induced from each. N tV. Riemannian metrics, then it holds d(P, qN)il:d(z(P), z(qN)) for any P, q'"GM. Hence. tv T-i(P)lldiameter of P. diameter of P=. -. So if we assume that P is not compact, then P is not bounded from Proposition 1. And hence Ptvis also not bounded.･ So just as the proof of Proposition 2, there. tv tv. exists a line c:(-oo, oo)-M. Then, by [1, Theorem 2, pp 120], M must be a Riemannian product of 1-dimensional Euclidean space Ri and a Riemannian mani-. tv tv. fold M i.e. M=RixM'. Hence Ricci curvature of M must not be positive at any point oftvM and hence Ricci curvature of M must not be positive at any point of. M) because -: fi--->M isalocal isometry. q.e.d. As a special case of the above Theorem, when dim M=2, we have Propotion 3. Let M be a 2-dimensional connected complete Riemannian manifold with non-negative Gaussian curvature, positive at some point of ltzZ Then P is a compact subset of Ml. Fourthermore if manifold M satisfies more additional conditions then we have. Theorem B. Let M be a 2-dimensional complete Riemannian manifold with non-negative Gaussian curvature K. If S.K dv, the total curvature of M] satisfies. S.K dv=2z (dv is the volume element of M) and KiO out side some compact set, then it may exist at most one pole in M i.e. the number #(P)=1.. Remarle. By a result by Cohn-Vossen, the total curvature i.K dv of a 211,. .. dimensional complete connected non-compact Riemannian manifold M with nonnegative Gaussian curvature satisfies O{!!S.K dvi:i2n. A typical example of a manifold which satisfies the conditions in the above theorem is a half cylinder. Pbtoof of the Tlheorem B. A manifold with a pole and satisfying the assumptions in Theorem B is non-compact. Thus by the classification theorem by Cohn-. Vossen, see [2;pp 438], Mis diffeomorphic to an Euclidean plane. And in this. case,weprovedthefollowingfactin[5]. . . : 1. i i 1.

(4) 4 M. MAEDA. For a point PeMl let A(P) be the set of all unit tangent vectors v at P which. tangent to a ray r: [O, oo)-M i.e. 7(O)=v. The set of all unit tangent vectors at p which we denote by tZH }(M) is a Riemannian manifold with the Riemannian. metric induced from the Riemannian inner product of Tp(M). Thus we can consider the Riemannian measure on 71b(M) and hence measure T;(M) =2z. By definition A(p) is a closed subset of 7;(M). Thus we can consider the measure of A(p). For any positive small e>O, let C be a compact subset of M satisfying '. '. 1. Sc K dv>=S. K dv-e.. Then for this C, we can find a compact set C,>C such that for any point qeM-C,,. ./. measure A(q),<=2z-S. K dv+e and for any ray T: [O, oo)-M starting from q, r([O, oo))nC==￠. The proof of this fact is done by using Toponogov's splitting theorem, see [1] and. [6;Lemma pp 17]. So if we choose C a compact subset including the support of the Gaussian curvature K, then Sa K dv=!M K dv. Thus we can choose e=O and we have a compact subset Co having the property mensioned above. In particular, measure A(q)=O for all qGCo.. , Now, suppose there exist two poles Pi and P2eM: Choose a point PGM-Co different from Pi and P2. Let ri: [O, oo)--->M and r2:[O, oo)-->M are rays such that ri(O)=Pi,i=1,2 and ri(d(Pi,P))=r2(d(P2, P))=P. Since ril[d(Pi, P), oo)=: 7i and. r2I[a(P2, P), oo)=: 72 are two rays starting from P, rnyi and 72 do not meet with. C, because of the construction of C. Then by 7iU72, Mis dicomposed into two mutually disjoint domaines whose boundaries are 7iu72. Let D be its one of. domains satisfying DcM-Co. Then, since D is simply connected and Gaussian curvature K on D is zero, D is isometric to a domain in'a flat Euclidean plane. E2 whose boundary is a union of two half lines (rays in ,E2). So the subset {vGZ"1(M):exp. tvGD for all Ost<r(p)}(=: Tl(D)) of Tb(M) is contained in A(P), where r(p) is the convexity radius of p. Hence measure A(P)lilmeasure 71b(D)=the angle between 7i(d(Pi, P)) and 72(d(p2, p)) measured on D>O. This is. acontradiction, q･g･d･ It is a very interesting problem weather the above theorem hold without the. assumption that S. K dv=2z or support of K is bounded (in this case we will need another assumption that K is positive or more weakly positive at some point). or dim M==2. ･. ' tt. t.

(5) A Note on the Set of Points which are Poles. 5. REFERENCES [1]. J.. Cheeger and D. Gromoll, The Splitting Theorem for Manifolds of nonnegative Ricci. [2]. J.. Cheeger and D. Gromoll, On the Structure of complete Manifolds of nonnegative. [3]. s.. Cohn-Vossen, KUrzeste Wege und TotalkrUmmung auf Flachen, Compositio Math.. [4]. D.. Gromoll, W. Klingenberg, W. Meyer, Riemannsche Geometrie im Grossen, 1968,. [5]. M. Maeda, A Geometric Significance of Total Curvature on Complete Open Surfaces,. Curvature, Jour. Diff. Geom. 6 (1971), 119-128.. Curvature, Ann. of Math. 96 (1972), 413-443. 2 (1935), 63-133.. :. Springer Verlag. Advanced Studies in Pure Math. 3 (1984). Geometry and Related Topics, Kinokuniya Tokyo and North Holland Publ. 451-458.. j. [6]. M. Maeda, Remarks on the Distribution of Rays, Sci. Rep. Yokohama National Univ. Sec. I, 15-21.. Department of Mathematics Faculty of Education. Yokohama National University. .. .. :.

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