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(1)A CERTAIN COMPACTNESS THEOREM Masao MAEDA g O. Introduction. In [8] , S.B.Myers proved the following fact which is known as the theorem of Myers : Fact 1 (S.B.Myers). Let M be a complete Riemannian manijbld whose Ricci curvature is. unijbrmly boundedfrom helow by apositive constant 1 >a thenforeach geodesic c: [O, n/ 4/Ji-. ] -M) there exists a conjugate point ofc (O? along c on c.. Thus each geodesic in M can not be a shortest geodesic between its end points if its length is greater than rr /VJ( and hence the diameter of M is not greater than n /vril.In this note, let all geodesics have arc-length as their parameter.. So we have Fact 2. Iet M be a complete Riemannian manijQ)ld whose Ricci curvature is unijbrmly boundedfrom below by a positive constant. T7ien M is compact.. Generalizations of this compactness theorem were given by many authors, see. W.Ambrose [1], E.Calabi [4], G.J.Galloway [6]. The methods used in these generalizations are based on the idea used by Myers which calculate the second variation of the length of a geodesic with length greater than n /VI and show that this second variation is negative by curvature assumption.. '. Another sort of generalization was given by K.Shiohama which is stated as follows;. Fact 3 (K.Shiohama [9] ). Let M be a complete Riemannian manijbld with positive sectional curvature Ko >Oforall tangent twoplane a.lfthe volume of M isfinite, then M is. compact. H.Wu also obtained the same result such as Fact 3, see [10].. Now the purpose of this note is to give a compactness theorem such as Fact 3. As a condition corresponds to the volume finiteness in Fact 3, we will use the condition which is derived by the fact known as Berger's lemma. This will be stated in the next section.. g1. Berger 's Lemmae In this note, let M denote a complete connected Riemannian manifold and d the distance function on M induced from the Riemannian metric of M. Tx(M) denotes the tangent space of M at x and Y(v,w) the angle between the vectors v and w.. Following interesting fact was found by M.Berger, see [7] ..

(2) 40. Masao MAEDA. Lemma A. Let M be a complete Riemannian manijbld and xEM the point such that the fanction py:M.R de:17ned by py(p) = d(p,y)forp EM has a relative maximum at x EM. T7ien for each non-zero tangent vector v E 71.(M), there exists a shortest geodesic 7 :[O,d(x,y)].M. from x to y satidying. k (v, 7(o)) f-g. T 2. This Lemma A was found by M.Berger under the assumption that M is non'negatively curved. But this assumption is not necessary as was pointed out by Y.Tsukamoto. If M is compact, then for any point y E M, the function py attains a maximum at some. point x E M. Thus we have Lemma A'. Let M be a complete Riemannian manijbld. Ilhen for each point yE M, there exists a point x E M satiofYing thefollowing condition :. (*) for any non-zero tangent vector vETx(M), there exists a shortest geodesic 7 :[O,d(x,y)]. .M.from x to y satidying. T 2. Y (v, 7(O)) $. Now, we will use the condition (") to derive the compactness of a complete Riemannian manifold. But only the condition (") does not derive the compactness. For example, all points of a complete flat cylinder Si(r) × Ri satisfies the condition (*) but. Si(r) x Ri is not compact. Here Si(r) is a circle wiih radius r in a 2"dimensional Euclidean space R2 and Ri a 1-dimensional Euclidean space.. Thus its needs more assumption to derive the compactness. Above example and many results stated before suggest that some curvature assumption is necessary. From these consideration, we have. Theorem. Let M be a complete Riemannian manijbld with positive sectional curvature Ko > O for all trangent two plane a. lffor each point yEM, there exists a point xEM, which is dUfflerentfrom y and satidying thefbllowing condition:. (*') for any non-zero tangent vector vE 73,(M), there exits a geodesic 7 :. [O, a] 'Mfrom x to y satiJi157ing. *(v, fr (O)) $. T 2. thenMis compact. ,. Remark. (i). The condition (") is weaker comparison with the condition ('). In("),. the shortestness for a geodesic 7 is not required. -.

(3) A CERTAIN COMPACTNESS THEOREM. 41. (ii). The assumption that for all point yEM the condition ('*) must be satisfied is necessary. In fact for any given small e>O, we can give an example of non'compact 2-. dimensional Riemannian manifold with positive Gaussian curvature and with the following property :. ,the Riemannian measure of the set B(M) :=. M' {y E M: for y E M,there exists a point x E M satisfying (*') }. is not greater than e.. For example, we consider the function f defined by. f(x)=( i-:il.ll-2 ･･･ x)o. l o ･-x<o fis a smooth function on(-oo, 1/2) andf"> O. Thus we have a rotation surface S in a 3dimensional Euclidean space R3 with xyz-axis whose axis of rotation is y'axis and whose. 1. generating line in xy-plane is given by the function y=f(x) [O,-i- ). This surface S is non-. compact complete and satisfies the following properties;. (i) Shasapolepo=(O,O,O)ES,namelyeachgeodesic7:[O,oo).Sstartingfrompoisa ray. Here 7 is called a ray if any subarc of 7 is a shortest connection between its. end points (ii) Gaussian curvature of S is positive on S. (iii) S-B(S) = (closure of (S-B(S)) is compact.. (i) isobvious.(ii)followsfromthepropertyf">O.Sincef'ismonotoneincreasing. 11. andf'(x)-. oo as x. - i-, we can find xo such that O<xo<7 and for all x, x ) xo. r(x) := J[)"C V'FiT7(Fl5-u) du > 2nx.. Then since the length of the geodesic circle Sr(x)(po) with radius r(x) centered at po in. S is 2Tx, we can see that for each point yESr(x)(po), there exists a geodesic loop 7:[O,oo)' S such that. (i) 7(O)=yand7([O,a])CB,(.)(po)(:=geodesicballwithradiusr(x)centeredatp.). '. (ii) 7 [O, g]and 7 [g,alare shortest geodesics (iii) 2nx >L(7) > O.

(4) Masao MAEDA. 42. Thus 7( g)=: x is a cut point of y along 7 [O, g] and the condition (*) is satisfied for point. x･ So B,(.o)(po)CB(S) and hence S ' B(S)cB,(.o)(po). For any e'>o, let h,f:R3. R3 be the homothetic transformation defined by he,(x)= e'x for x G R3 where x E R3 is. - Let Se, := h,,(S). Then S,,is a complete nonidentified with the position vector ox. compact surface with positive Gaussian curvature and B(S.,)=h.,(B(S)) because h., is a affine transformation. Thus for any small e >O, we can choose sufficiently small E' > O. such that measure of Se r B(S,,) < e.So M= Se,is a surface as required. For the proof of the theorem, we need some preparations. Following facts are obtained. by J.Cheeger and D.Gromoll in [5].. Let M be an n -dimensional complete non-compact Riemannian manifold with nonnegative sectional curvature. Then there exists a family of totally convex subsetslCt1t )o of M which satisfies the following properties;. (1) C,CC,, if t<t' and C, =l q E Ct,:d(q ,aC,) ]l; t' 'tl. (2) UC,=M t)o. Here a subset CCM is called to be totally convex if for any two points q,q' E C and any geodesic 7 :[O, B]'M from q to q', it holds 7 ([O, P ])C C.. Totally convex subset CCM has a structure of an m-dimensional (m ;S n) totally geodesic submanifold of M as its interior and posibly a boundary aC. So we can denote. as dim C= m. Boundary aC is a topological (m-1)- dimensional manifold. If aC4 ￠, then the set. Ct :=l q E C:d(q ,aC)-2}rtl is also a totally convex subset for each t)O. In particular, if we put 0 := 'qMEa aX c d(q, OC),. then Ca is a totally convex subset and dim C6 < dim C. Let denote Cmax := Cs･ Now for the family of totally convex subsets (Ctl, if %; ￠,let put. (Co)max'= Ci. Then Ciis totally convex and dim Ci<dim %.If C'I ￠ ,then let. Ci. := C2 max.

(5) A CERTAIN COMPACTNESS THEOREM. 43. C2 is also totally convex and dim C2< dim Ci. Continuing this method, we have a sequence of totally convex subsets. Co )Ci)c2)c3 ･･･)c' such that aCil ￠.Thus we get a complete totally geodesic submanifold C' of M with aC'= ￠.C'is called a soul of M. If dim C'> O, then the sectional curvature Ko = O for all. '. tangent two plane o which is spanned by two vectors v and w such that v is tangent to Ci and w is orthogonal to the tangent space of Ci. So if the sectional curvature Ko of M is positive for all tangent two plane 6, then dim C'= O and hence C' is a one point set.. Proof of the theorem.. We assume that M is non'compact and will derive a contradiction. Let ICtlt)oand ICil,.1,2,...,l are the families of totally convex subsets stated as above. Since Ko>O for. all tangent two plane o ,we know that dim C'=O and hence Ci=(yl for some point yE M.. ' For this point y E M , the condition (*") must be satisfied also. Thus there exists a point x E M such that for any non-zero tangent vector v E Tx(M), there exists a geodesic 7 :[O,. a].M from x to y satisfying Y (v, fr(o)) $z12.For this x, we can consider the following two cases;. Case (I). xXCo or xE aCo. In this case, there exists tol O such that x E aCto.This follows from the property (2).. Case (II). x E Co and xXaCo. In this case, we can find a number lo such that xEClo and xX aClo. So putting to :=. d(x,aclo), we have xE ocloto= {q E aclo:d(q, aclo)litol and in particular xE. aciot.･ ' We will prove the theorem in case (II). If case (I) occur, then we can prove the theorem similarly by replacing CIOto in case (II) with Cto･. Let k := dim cloto and {xi}ccloto be a sequence such that. xi X acloto for i=1,2,"'. and x.× as i.oo. From a property of totally convex subset, interior of Cloto has a structure of k-dimensional totally geodesic submanifold of M. Thus at each point xi E interior of cloto, we can consider the tangent space Txi(cloto) of cloto which is a k-. dimensional subspace of Txi(M). Then Txi(CIOto) also tends to a k-dimensional subspace in Tx(M) as i.oo.We denote it as,Tx(CIOto)C Tx(M)..

(6) 44. Masao MAEDA. Let Cx :=(a7(O):7 :[O, e1-･CIOto be a geodesic in M starting from x and a E R, a )O}.. Since Cloto is totally geodesic, Cx cTx(Cloto) makes a k- dimensional cone in Tx(M).. Cx is called a tangent cone at x. Thus there exists a non-zero tangent vector voE. T. (Clot.) such that C. c(wET.(CIOt.):Y (v.,w ) lll - li- lUlOl. For this vo, the condition (') must be satisfied also. So there exists a geodesic 7 : [O, P] --+M from x to y. satisfying Y(vo,7(O))$L. On the other hand, by t2otal convexity, we have 7([O,B]) GCIOtoi.e.7(O) E Cx.Thus. TT. Y (v.,7(O)) ) - i' and hence we have Y (v.,7(O)) = z}'.So, 7(O) E aCx.. Nowsince7([o,p])cclot.and7(o)=xEaclot.,7(B)=yEintclot,(=clotiaclot.), we know 7 ([O,B])c int Cloto .This follows from a property of totally convex set, see. [5] . 7 ([o,B])c int clot means 7 (O) E Cx and this is a contradiction.. o. Q.E.D.. References [1] W.Ambrose, A theorem of Myers, Duke Math. J. 24 (1957) 345-348.. [2] A.Avez, Riemannian manifolds with non-negative Ricci curvature, Duke Math.J.,39 (1972) 55-64. [3] M.Berger, Sur quelques variet6s Riemanniennes suffisament pincees, Bull.Soc. Math. France, 88 (1960) 57-71. [4] E.Calabi, On Ricci curvature and geodesics, Duke Math. J. 34 (1967) 667-676.. [5] J.Cheeger and D.Gromoll, On the structure of complete manifolds of non-negative curvature, Ann. of Math., 96 (1972) 413'443.. [6] G.J.Galloway, A generalization of Myers theorem and an application to Relativistic. cosmology, JDiff. Geom., 14 (1979) 105-116.. [7] D.Gromoll, W.Klingenberg and W.Meyer, Riemannche Geometrie im Grossen, Springer-Verlag, 1968. [8] S.B.Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941) 401-404. [9] K.Shiphama, An extension of a theorem of Myers, J.Math. Soc. Japan, Vol.27, No.4 (1975) 561-569. [10] H.Wu, A structure theorem for complete non'compact hypersurfaces of nonnegative curvature, Bull. A.M.S., 77 (1971) 1070-1071.. [11] S.Yau, Non-existance of continuous convex functions on certain Riemannian manifolds, Math. Ann., 207 (1974) 269-270.. Department of Mathematics Faculty of Education. Yokohama National University.

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