- PERELMAN'S FORMALISM IN POTENTIALLY INFINITE DIMENSIONS 425
LEMMA 8.41.(1) 900 -m = 900 - - 2of - - -f +^0 (N-1)
OT T
= N + R - 2 ° f - L + 0 (N-^1 ) '
2T OT T( 2) giQ = - z:i + 0 ( N-1) '
(3) gfj = gij + 0 (N-^1 ) = 9ij + 0 (N-^1 ),
(4) g~(3 = ( 1 - ~) !Jcx(3,
(5) g: = 0,
(6) g~ = o,
where ; 7 9ij = 2Rij·
PROOF. Let 7 = 7 (x, T) ~ ( 1-~) T. In the formulas below, 0 denotes
the time index; i, j, k, ... denote indices on M; a, /3, '"'(, ... denote indices on
SN; and a, b, c, ... denote arbitrary indices. The pulled-back metric is given
by
g1;J (x, y, T) = ~~: (x, y, T) ~~: (x, y, T) !Jed (0 (x, y, T)),
where z = x, y, or T. Using the formulas for !Jij, !Jcxf3, !Joo, and !Jicx = !Jio =
!Jcxo = 0, we obtain the following.
(1)g 00 (x,y,T) = (
0
!0
(x,y,T))2
!Joo('P(x,y,T))(
0-)2
=
0~ !Joo (x, y, f)= (l-2f _ 2Tof)
2
(N (l-2f)-l +R)
N NOT 2T N( (
2f 2T Of) -2 )
= 1+2 -N - NOT + 0 (N )x (R+~ (l+~)+o(N-^1 ))=R+-N ( 1+-2f) +-2 N ( -----2f 2T of) +O(N -1 )
2T N 2T N NOT
N f of -1
=R+----2-+0(N )
2T T OT= 900 - (. x' y' T ) - - -f 2-;:,;-^0 f + 0 ( N -1).
T UT