Pinching Implies Strong Pinching 125

cannot be expected. (4) implies (1) immediately: tracer = S<oû)\ co"> = i X'^w;i Z^i>«

V V

= i zl'f^kijii^ik^ji ~ ^ii^jk) = Z ^kiik-

кФ1

DefineRo { X , Y) Z: =i(J +^) «7, Z> Z-<X, Z> F) and D:=R-Ro. SinceD and i? have the same symmetries we have from (7), (8), (9) in [2] for unit vectors in Mpi

KD { X , Y ) Y , Uy\^2 - 'f' { A^ôl «И^-^) if XIU) (5)

If Z1Z, У then |<D (X, У) Z, l/>| < f (J - ^) (6)

\m = max |<D (JC, У) Z, l/>| < (34/36)^/^ (^ - 5). (7)

Remark . In forthcoming papers on the differentiable pinching problem Sugimoto and Shiohama have used ||Z)|| ^A: (with normalization J +^ =2) as a pinching tion. IIDIK/: clearly implies for the sectional curvatures 1—k^K^l+k. The converse is not true since for the complex projective space \\D\\ =jk. However l-k^K^l-bk and (7) imply \\D\\^{34/9y^^ k. We rewrite (4) as

( qco , œy = A+ô-hiY.' ^kijiCOijCOki (8)

and apply Schwarz' inequality (note |ш| = 1):

\iTD , tj , œ , jœ „ \^Q : 'D'^j , Y'\ (9)

( Schwarz' inequality can be applied in various ways to (8) leading to different pressions for the [ ]-bracket in (2); in our computations they were all of the order ^3/2 Qj. ^oj-se If ^2) is a poor estimate the loss probably occurs in (9) since the following estimates seem fairly sharp to us.)

TDbj , = I Dlj,+ X Dij,+ X Dlj, (10)

k , l = j,i k,l^i,j k,l,i,J

We have from (5) with /, / fixed and a^: ^D^jjiCZk J^ljjd'^'^

i / Vi^

2 - ''' { A - ^ô ) ^Y^a , D , J=^ [ Y^DljA . (11)

к \ \k /

If /#/,7 we have from (6) with a,: =i),,^,(5], /^â,,)"'^'

f ( J - ^ ) ^ Т^а,В,гА = (^В1,Л . (12)

к \ \k /