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(1)The Injectivity Radius of Certain Manifolds By. Masao MAEDA" Let M be an even dimensional compact simply connected Riemannian manifold with the sectional curvature Kb satisfying O<Kb$constant 2 for all tangent. two plane a. Then, well known W. Klingenberg's theorem states that. i(x)2-fz- forallxEM. '. Here i(x) is the distance from x to its cut locus C(x) in M. This estimation for the injectivity radius of the exponential mapping exp.: T.(M)->M does not hold in general without the assumption that M is of even dimensional. M, Berger showed that on 3-dimensional sphere S3, there exists a homogeneous Riemannian metric g such that O<Kb:$R and there exists a point xoE(S3, g) which satisfies. i(xo)<iR-. And more generally in [3], M. Tani showed that on any gdd ' ･･･, there exists a Riemannian metric g such dimensional sphere S2n"', n==1,2,. ' i(x')< vT2-. Thus in odd that O<KbslR and a point xX(!(S2""', g) satisfying dimensional case we need more assumptioB to get such estimation. Berger's example shows homogenity does not suffice. Noticing these facts, in this note, we will show the following:. THEoREM. Let M be an n (l4)-dimensional Riemannian manifold dij7`bomorPhic to Sn and its Riemannian metric conformally equivalent to a standard metric of Sn. 111e its scalar curvatztre is Positive and the sectional curvature satishes Kb:.{R for all two Plane a. Then. i(x)llllvT2-- forallPoint xEM. For the proof of this fact, we use the results obtained by H.B. Lawson and J. Simons in [2]. See [2] forthedetailed facts. Let X be a smooth vector. field on M and gt:M->M; tER the 1-parameter group of diffeomorphisms generated by .X. Let S be a P-dimensional compact Riemannian submanifold of M and V(t) the volume of thesubmanifold gt(S) with respect to theinduced metric, i.e. V(t):==S.･N/r(Z58Zi5(eci J-gtg)(8.,8.)dvs. Here g is the Riemannian metric of. S, dvs the canonical measure induced from the Riemannian metric g and 6:S-APT(M) an dvs-measurable section of the P-th exterior product of the ･"M. tangent bundle of M with the property that for dvs-almost all xGS, g. is a " Department of Mathematics, Faculty of Education, Yokohama National University..

(2) 2 M. MAEDA. simple vector of unit length which represents T.(S). Then, from [2: Theorem 1, pp. 435],. ' ' d2 <1) dt2 V(t)lt-o=S,{-<u4X(g), g>2+<u4x,.4x(g), g> '. -+IIuiZi(6)II2+<7x,."X,<'>}dvs. where u4i and Vx,oX are defined as follows. Let 7 be the Riemannian connection of M and (-jZX:T.(M)-.T.(M) be defined as u4i( Y):--7yX for YE T.(M).. ?. This map can be extended uniquely as a derivation on APT.(M). Namely, for a P-vector YiA ･-･AY,EAPT.(M), u4X(YiA ･･･ AYp):= ]ll] YiA ･･･ AYi-i. i=1 Au4X(Yi)A Yi+iA ･･･ A Y,. Let 7x,oX:T.<M)-T.(M) be alinear map defined as. 7x,yX:=7xVyNXmV7.gX for YET.(M). where 9 is an extension of Y as a local field. And this definition does not dde erPievna2ioOnn. oannYApeE2j iSj'.On Of Y' And aiso 7x7oX is extended uniqueiy as a. ' Now, foramoment, weassumethat Mis diffeomorphic to an n-dimensional Euclidean space E" and g:M-->En is the diffeomorphism. Let {zL`} be the canonical orthonormal coordinate in Enand xi:=u`og a coordinate in M.. Assume that M has the Riemannian metric expressed as gij(x)dxidxj= e2f(X)5ijdxidxj for some smooth function f, where Sij i's the Kronecker's delta･. We use the Einstein's convention. Putting f,:=O.f70xZ, we have. gij=e"2fSij, <2). { ].Zle }==LSk +f,S}-f,5,h. ' Put fij:=OiL･-fifj where g,,gjk=6,le and {7･Zle} is the Christoffel's symbol. + ± fkfk6ij. Then the curvature tensor is expressed as. '. (3) R,,,i=-O Si?.}6{,/k･}+{,6,}{z･z}-{ji.}{:･z}. '. Then, from (2) and (3), we have. --. (4) R,,,e==f,,6S･-h,SS･+5,,fS･-6,,ft･. In particular, when i7SL we have. s. (5) Rijij=Rijileglej=e2fRijile6kj=e2f(fii+fjD. Now assume that M has the positive scalar curvature. Then, from (5). O< - ]X 2 2-4fR,j,j i=1 J'#t. n･. == - 2(n - 1)e-2f ]2)n fii. i=1. '.

(3) The lnjectivity Radius of Certain Manifolds '3 ' ' '. =-2(n-1)e-2f{t?.,O,f,+(g-1)t?.,f3･}･. So, we have '. <6) ￡.,O,fi<-(g-i)t/i.i).,f3･･ '' ' O For vector fields X.,:== ox., a==1,2,･･･,n on M, let {gg}tER be the 1-. parameter groups of diffeomorphisms of M which generate X.. V.(t) denotes the volume of g9(S). And we formally calculate.E.l}, ddt22 V.(t)lt=, by using (1).. Hereafterweusetheindexnotationsuchthat . ISh,l,m,a$n, ISi,j-s{P, IS.k-<.q,P+q=n. For each point xEMI {Xi(x),･･･,i¥.(x)} is an orthogonal basis of T.(M) with same norm ef. So let {ei,･･･,ep,ni,･･･,nq},P+q=n be an orthonormal basis of T.(M) such that eiA ･･･ Ae. =6.. Then {eb ･･･,ep, ni, ･･･, n,} is obtained. from {Xi(x), ･･･,X.(x)} by some orthogonal transformation {aM} and the function e-f. That is. et= e-fal X.(x), nle=e-faifX.(x).. ' Since I[u4Xdi(6.)ii2=<cyjZict(8.),e.>2+Z<(YZXdi(ej),nle>2, the integrand of ' j''ic. d2. liii dt2 Vcr(t)[t-o. '. == :iP{-<u4Xat(e.), 6.>2+<u4Xdi,-.jZitu(e.), e.> +H,-jZidi(6.)H2+ <7..,e.X., g.>} = lll] <aidiu4iat(6.), 6.> +.l;,1,,<u4Xat(e,), nk>2+ ]{) <7x.,#.X. 8.> =2 ￡ <u4Xat(e,), e,><UZXct(e,･), ej> + 2,<u4iou.jZiat(ei), e,>. ai;j'- a,z. ' +.lllLl,,<uiZiat(ej').,nfe>2+.Z,,,<Vx.･,eJ'Xooej'>'. =:I+ ll +M+IV. We will calculate these four terms respectively. Since. '. <u4ict(ei), ei>= ;l.l] al･aY･ { IM. }= IIIi) ai･a: (ft6ge +flt6Y-f.5la) = >i) aS･ag･f},==.fL,,. ' ' ' I=:il),1;j<u4Xct(ei),ei><u4Xat(ei),ej>=p(p-1)?fz. .. NeXt, we calculate the second term. The following can easily be verified. <u4iat..fziat(et), ei> =;aS･ath･ { IM. }'{ .4cr }. =,;l,),a!-a:･L(feop+f.5ge-f.6t.)(f.S2+f.Sin-A6..). ==2alaSfefh-aga?ftA+fZ-a2a,ct･¥fZ..

(4) 4 M. MAEDA So we have. ll = 2)<u4Xct.iZXat(e,), ei>=Z(2-'n)ag･alfi.fl,,. a,i i. =￡(2-n)(aS･f,)2.. i. Using 2a}aZ =･5jle==O, we have. L. <u4iat(e,), nle>2 = :;;p(aSaw{ l"crZ })2 .. ==]E)(aS･aW(ft6gy+fh,Sz't-.71.5i.))2. m =(aS･a2 ft - aS aYL,,)2 .. u.. Hence z <u4Xct･(e,),,n,>2= Z {(aSia2fi)2+(a,at･ aYL.)2-2aS-a,at･ aZ djfiL.}. a,ik. a,Jl le == Z {(aS･aff,)2+(agy･ ayL.)2} a,1',le == ￡ (a2)2(aS･f,)2+ 2 (a?･ )2(a7fl.)2. cr,J',k. a,j',le. =li](7Z-P)(aSft)2+l:I]p(a7,nfin)2.. Since {aM is an orthogonal matrix, :i)(aZhft)2==Z.(ai.ft)2-:i,](aS･fe)2=T-:X]fZ-l;,)(aS･fi)2･. so. M =(n-P)Z(aSf,)2+P￡(aYf,,)2 =(n-2P)Z(a}f,)2+PZfZ. By definition the fourth term becomes. J'k j'a. N == 2.<Vi.,,ejXcr, ej>. a,j. ==.i,.a;a?L -oa.. { ln. } = Z a;･aY(O.,f},Si'n+5.,fiSW-O..1`}.St.). a,J',m. -peo.fh<-p(-,"--i)pfz.. ,. Here we have used the assumption (6). From the above calculation, 1I] ddt, V.(t)I,=,-S,(I + ll +m+Iv)dv,. <S,{P(P-1)7fZ+(2-n)¥(a!f,)2 +(n-2p)¥(a!･ft)2+P¥fZ-P( Z -1)?fZ}dvs =S.{P(P+i-ll-)?fZ+2(i-p)¥(a:･f,)2}dv,.. ･!,.

(5) The. Injectivity Radius of Certain Manifolds. 5. Proof of the Theorem. We assume that there exists 'a point yGM such that i(y)< .v:- and derive. a contradiction. Let yNEMbe apointsuehthat i(yN)==Min{i(x):xGM}. Then, by [1; Lemma 2, pp. 226], there exists a normal closed geodesic c:[O, 2i(yN)].M such that c(O)=c(2i(yN))=yN. Since c([O, 2i(Y)]) is a 1-dimensional compact sub,. manifold of M] we can apply the above argument. Choose a point zGM -c([O, 2i(yN)]) and let U.(z)cM-c([O, 2i(Y)]) be an open convex ball with radius. r>O centerd at z. Using a stereographic projection and the diffeomorphic map c. from M to Sn, we obtain a diffeomorphic map g:M-{2}->En which is conformaL Let ut be an orthogonal coordinate of E" and xt:=uZog be a coordinate in M. For a positive number r',O<ri<r, let ￠ be a smooth function. ' on M such that O$￠S.1 and ･,={ g･･ g.n bK-,,Y,z`g'. And consider the vector fields Xcr, .=i,2,.,,,. on M given by. xa,=l￠o2a on M-{z}. tO atz. Let {g2}tER be the 1-parameter group of diffeomorphisms on M which generate Xa. Then from the construction of ￠, our previous calculation can be applicable. to this case. And hence we have, putting P==1, :i; S,2, v.(t)s,=,<S,{(2-ll-)]Ipfz}ds:so. where ds is the volume element of c. Thus we can find a vector field X" on M, not vanishing on c, such that. dt2d2V.(t)I,=,<o. Then, by the same argument used by W. Klingenberg to get the estimation of the injectivity radius for a compact orientable even-dimensional manifold with. the sectional curvature O<K.$2, we can easily get a contradiction, see [1;. Satz, pp. 227]. q. e. d. References. [1] D. GRoMoLL, W. KLiNGENBERG and W. MEyER, Riemannsche Geometrie im [2]. Grossen, Lecture Note, (1968) Springer-Verlag. H.B. LAwsoN and J. SiMoNs, On stable currents and their application to global problems in real and complex geometry, Ann. of Math. 98 (1973), 427-450.. [3] M. TANi, An example of an odd dimensional positively pinched Riemannian manifold, T6hoku Math. Journ. 21 (1969), 532-538..

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