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(1)A Note on Riemannian Pinching By. Yosio MUTO DePartment of Mathematics. (Received Aug. 11, 1966) . In the present paper some properties of the sectional curvature of a com-. pact complete Riemannian manifold are studied. Some numbers are assumed to be given some of which are connected with the sectional curvature or the scalar curvature at a point and others are connected with the covariant derivative of the curvature tensor.. ' g1. Let M be a compact complete Riemannian manifold of dimension n. where the sectional curvature k at a point P satisfies ' L(P) ;;l le ;ll K(P),. '. L(P) and K(P) being positive numbers depending on P. Let Knjih denote the curvature tensor of M, and 7k the operator of covariant .d.ifferentiation with respec't to the Christoffel e}. We use natural frames. Indices run as follows: h, i, 1', le,l== 1, ･･･,n.. Sipce M is compact, M admits a positive number D satisfying D ).IZ`X ic YjYiXhVzKicjih1. for every orthonormal vectors X, Y and every unit vector Z at every point of M.. Then we have N. PRoposiTIoN 1.1. Let P and Q be any pair of Points of M with distance. d(P,Q)=l. Then. (Ll) , K(Q)SK(P)+Dl,. (1.2) L(Q) ;-;}i L(P)-Dl.. ' PRooF. Let r be a geodesic such that r(O)==P, r(l)= Q. The tangent vector of r is denoted by Z(s). Let the sectional curvature of M at the point Q in the plane direction a spanned by orthonormal vectors Xl2, YQ be denoted bxYa)k laxQ6?'y(Ilg =W, 9Q,COwneSidgeert Parailei vector fieids X(s), y(s) aiong r such that. '. --dds- Kiryyx == Z`XkYjY. iXh7tKkjih.

(2) 30 . Y. MuT6 along r, hence. d --zz'g"Kxyyx sD. This gives Ik(a, Q)-k(a,, P)lS DI, where a, is the plane direction spanned by X(O), Y(O). From this inequality we obtain (1.1), (1.2) immediately. Let the first conjugate point of P. along the geodesic r considered above be denoted by r(s,), s,>O. We define tsi by. Yi(Si)=O, Yi(S)>O if O<S<Si, where yi(s) is the solution of the differential equation. o. (1･3) SiY,-+(L(P)-Ds)y=O,y(O)=O,yt(o)=1.. Then we have s,.<.si by virtue of the Bonnet-Rauch ' comparison theofem [1]. s, is a function of L(P) and D. When L(P) =1, this function will be. a. denoted by f(D). f(D) is defined only over some interval L. If we put ' ' t= ･viL(P) s, (1.3) is transformed into. d2y. tt. (L4) . -d-l2-+(1-Ct)y=O where C is given by. --3J. (1.5) C=(L(P)) 2D.. HenceN/L(P)s,isequaltof(C)andwebbtain . ;':'"' '''. (1.6) s,=L(P)M'211e(C).s- ttt 1 ''. '. '. Aswehaves,$s,,wefindthatMisS-pinchedwere6satisfies ,J. ". L(P)-Ds, 6> K(p)+D--si' Substituting (1.6) into this inequality we get. a. L(P)(1-cr(C)). (1 .7) S> -K(p)+L-(tt)-cf(c)". , -1. ･ find If C is sufficiently small, f(C) can be obtai'ned as a power series. To. this series we only need to substitute ' into (1.4).. L. . y==sint+Cq,(t)+C2g,(t)+･･････ We get 9i"+9i-tsint=O, {D, ii + 9P2 - t9Pi = O,. ".t.-t------.-----'. I. I. l. l. i l.

(3) ANote on Riemannian Pinching 31 '. '. ' 9i(t)=-i}t2cost+t'tsint, . .' -------------te"--.. Assuming C to be small we can put f(C)==z+a which must satisfy sin (z+a) +cg),(T+a) +c2sD,(z+a) + ･･････ .= o.. Fromthisequationweget ' a - -T4- ftc+ -･･-, hence. '. ' Z2 4 C+ ･-･-.. (1.8) f(C)=z+. Substitutingthisresultinto(1.7)weget ' lm LZvD-L..t( LTvD-)2- ....... (i.g) 6> s+ ,TvDI+t( ,ZvDI)2+ ･･････' where we have dropped P in L(P).. Thus we have proved the THEoREM 1.2. Let D be a Positive number and M be a comPact comPlete RieMannian manijCold where the curvature tensor satisfies. lZiX' te YdYiXh7,K),,j,,I g D･. .t. for every unit vectorZ andfor every orthonormalvect'ors X, Y. if the sectional curvature fe of M at a Point P satisfies. L f{g-k f{; K. ' t. Positive number L, K and if C== L-Lg-D .t belongs t for.se?zpe to the interval I over which f(C) is defined, Mis 6-pinched with 6 satisfying (1.7). The function '. f(C) in (1.7) can be expanded in a Power series (1.8) of C for szoj7ciently small C and in this case we get (1.9).. We 6an find out the graph of "the function f and the least upper bound ?gugihbeYstinmUaMtieorACaoif sCa. iCuiations･ But we have another method which gives. LetPb'e-apointofMandRpbeanumbersuchthat ''. (1.10) '' L(P)-DR.>O, tt･VL(P)-DR.=T.' '"' ' Thencomparing(1.3)withthedifferentialequation , , -ddZY, +(L(p)-DR.)y=o '. whichhasasolutionsatisfying ' '.'.

(4) 32 ･ Y.MuTO y(O) ==y(R.) =O,. y(s)>O for O<s<Rp,. weobtainsi<Rp･ / -･,' ･', If we put ･6 == L-iDIRp, . (1.10)isequivalentto ･ , ･ ,- /.i,,.,' ･'･ f . 1-g>O,g2T63==z2L-3D2.. e ec2ise63. tAhS. o?aS a relatiVe MaXimum 4/27 at 6 =2/3, weionly need to consider. ' (1.11) n2L-3D2 :$ 4/27. If the least positive number6satisfying '. e2-63=z2L-3D2 '(:g4/27) -･tt isdenotedbyrc,wehavercS2/3and . ･,･. o. -. H6nce we get tSi<Rp=D'iLrc･ tt (1.12) '.., ･6> Kl-rc ･.,. ･.t. L ' T+rc'. t .t t Thuswehaveprovedthe ･'･'････:'t'''･･･. THEoREM 1.3. Let Mbe a comPact comPlete Rie'mannian manilfold and D,L;K be numbers givn in Theorem 1.2. 11f L and Z) satisyly (1,11) and rc is the smallest Positive root of x3--x2+z2LH3D2= O, then '`M is' '6-Pin'ched ivhere 6. satisfies(1,12). . :.-' ,,. ., From (1.5) and (1.11) we find that, if C satigfies , . ,,,, C53im3ap, ". .'i. thereExeAXMiSSSLEa fiIU.M?ferKf,//;26, 3;,,,Hesn.c,g stuhc.ht C belopgs to L. --. -''''1･ ' ･K±1,,L==-S-,''b!' sl.VIiioM-,l '''", a. tt we get rc =2/5 and 6> 1/4. If moreover M is simply connected, M is hQmeo'. morphicwithasphereSn. ',/ ,.. ･,t-. .'.., t/ tt ttt. g2. Let M be a compact complete Riemannian manifold of dimension n ll;3 and K be its scalar curvature K== Kkj,eg,igkh. We define Ste.iih by. (2･1) Slej'ih =Kte,pmv- n(nK-v (gkhgdi-g.ihgne)'. Then we have. ･ stej･ihgp'igkh=,o. '. FromBianchi'ssecondidentityweget ',,. 1.

(5) ANote on Riemannian Pinching 33 (2･2) n(nTl)(7iSkjih+7kSjiih+7jSileih)gjigkh+(n=2)(n-1)viK=O.. We define S(X, Y) by S(X, Y)=SkjihXicYjYZXh wh3n X, Y are orthonormal vectors. We put. (2.3) S:MaxlS(X, Y)1. Let Kp be the scalar curvature at apointPof M. Then it will be easily. seen that the sectional curvature k(o, P) satisfies ' 1. (2･4) IT('.-1)L Kp-SS le(a･ P) S n(nl-1) Ki'+S' Let T be a positive number satisfying. '. (2.5) IXi`X2kX3jX42Xshl7iSkjzhl;IEIT for all unit vectors X,, X,, X,, X,, X, and at all points of M. The we get. (2.6) lv.Kl ;.:{ -n(ge-=i)l-S<32n pt ?-)-. where X is an arbitrary unit vector. If Q is a point of M with d(P, Q)== l, the sectional curvature fe(a, Q) of M at Q satisfies (2.7). 1. n(n - 1). nm2 3n-2 Tl ;S k(o, (?) K.-S-. :ll n(nl-1) KP+S+ 3."i22mTl by virture of (2.4) and (2.6).. We put ･1 A= n(n"1) Kp-S, (2.8) L. 3n-2 B.= n-2 T. and assume that there exist positive numbers r satisfying '. (2.9) A-Br>O, r2(A-Br)=T2, Let r, be the smallest one. Then we obtain the inequality d(P, Q)<r, after the same reasoning as the one carried out in S 1.. If we put p,= A-iBr,, p, must satisfy. (2.10) 1-p,>O, p,2-p,3==n2A-3B2. p, exists only when z2A-3B2:S4/27 and satisfies O<p,:$2/3. From (2.7) and. l<B-iAp, we get the THEoREM 2.1. Let Kl) be the scalar curvature at a Point P of M and let.

(6) 34 ･ Y.MuTO. I. S and T b2 defined resPectively by (2.3) and (2.5), where Sk,･ih is given by (2.1).. l. 1[le T satis.17es T2Am3B2:$4/27 where A,B are de.fined by (2.8) and p, is the. smallest Positive root of x3-x2+z2A-3B2==O, then M is 6-Pinched where S satisfies. ' {Ki)---n(n-1)S}(1-p,). (2･11),., . S>K.+n(n-1)s+{Kp--n(n-1)S}po' ExAMpLE 2.L If n==3 K.==3os, T=-2;Vi;,,2s'gH, 8. then 6>1/4. If moreover M is simply connected, M is homeomorphic with a sphere S3.. ExAMpLE 2.2, If K>n(n-1)S and T2A'3B2:{- 4/27, Mhas positive sectional. ". curvature.. Reference 1) II,E. RAucH: Geodesics, symmetric spaces and differential geometry in the' 'large.. Comment. Math. Helv., 27 (1953),294-320. ･ ,. .. ".

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