432 8. APPLICATIONS OF THE REDUCED DISTANCE
so that
.. n p q
A - k£ 8xi 8xJ ~ 81/Jr 81/Jr
R = g 8,,Pk 8,,P£ L...J (Rpq + V' P V' qf) 8xi 8xj
r r p,q=l
- (i -~r
1
g: (Re ( sN) .μ - Tg.μ ~\7 f I' + Tg~flf) mod O(N-^1 )2 ( 21)-1 gr43 N -1 -1
= 2~1 - IV'll + R + 1-N ----;-2N9af3 mod O(N )= 2~1 - i\i 112 + R + ! (1 +
21) (N - 1) mod O(N-^1 )
2r N= ~ (N -
1
+ r(2~1 - IV' 112 +R)+1) mod O(N-^1 ).
r 2This is the same as the formula on p. 13 of [297] except for the -1 in N;-^1.
If we instead choose the metric g on sN so that Ra13(SN) = !9a(3, then we
would obtain the exact formula.
The integrand for the entropy W (g, 1, r) is related to the scalar curva-
ture of the hypersurface (He, gmlHJ by the following formula:
r(2~1-:IV'll^2 +R)+1-n=rR-n-N;l mod O(N-^1 ).
6. N ates and commentary
Section 2. Corollary 8.17 is from §7.1 of [297].
Section 3. Theorem 8.26 is from §7.3 of [297]. A localized version of
the theorem is given by Perelman in §8.2 of [297].
Section 4. Theorem 8.26 is from §11.2 of [297].