- PERELMAN'S FORMALISM IN POTENTIALLY INFINITE DIMENSIONS 421
LEMMA 8.40 (O'Neill). Let X, Y, Z be vector fields on B and let U, V, W
be vector fields on F. Then(1) R(U, V)W = K(U,V)W - JGJ~ (9F(U, W)V - 9p(V, W)U),
(2) R(X, V)Y = -1(VxG, Y)V = -1Hess 9 (!) (X, Y)V,
(3) R(X, Y)V = R(V, W)X = o,
(4) R(X, V)W = R(X, W)V = f(V, W)VxG,(5) R(X, Y)Z is the same on either B or M.
Here, K denotes the curvature tensor of (F, 9F) and
1
G = grad f = r;:; ( N ) Oy ·
2yT R+ 2TLet {Ba}~=l be a basis of tangent vectors on SN and 9a(3 ~ 9 (Ba, Bf3)·
By (3) we have
R(8i,8j)ea = O, R(8i,8T)ea = O, R(Ba,ef3)8i = O, R(Ba,ef3)8T = O,and hence
( R( oi, 8j )Ba, ok) = 0, ( R( oi, 8j )Ba, 8T) = 0, ( R( oi, Oj )ea, ef3) = 0,(il(oi,8T)ea,8j) =O, (il(ai,8T)ea,8T) =0, (il(oi,8T)ea,ef3) =O,
( R(Ba, ef3)8i, e 7 ) = o, ( R(Ba, ef3)8n e 7 ) = o.
By (1) we haveR(Ba, ef3)B 7 = K(Ba, ef3)e 7 - JGJ~ [9(ea, e 7 )ef3 - 9(ef3, B 7 )Ba],
and hence
( R( ea, e(3)e7, Bo) g = T (K( ea, e(3)e7, Bo) g
1- ( N) [9(Ba, B 7 )9(Bf3, Bo) - 9(Bf3, B 7 )9(Ba, Bo)]
4 R+ 2T
TR
2 N ( R + !Jr) (9(37~ao - 9a79(3o)
= 0 mod O(N-^1 ),
since JGI~ = 4T(R~~), g (e(3, e 0 ) = r9 (ef3, Bo), and
1 '
(K(Ba,Bf3)B 7 ,Bo) 9 =
2N (9(3 7 9ao - 9a79(3o) ·