1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. PERELMAN'S FORMALISM IN POTENTIALLY INFINITE DIMENSIONS 429

Using (6.15), we find that

goo= !Joo - J\7fJ

2

= ~ ( ~ - [r(2~f - J'VJJ

2

+ R) + f ~ n]) + 0 (N-^1 ),

g~f3 = 9m ( w (oa), w (8{3)) ~ 9m (Bai af3) = 9~{3(1/Jr(x), y, r)

= (i-

2

f;\I!)!Ja{3(1/Jr(x),y,r),

gffi = 9m(w* (8i) 'w* (8r)) = 9m((1/Jr)* (8i) ;or)

= -\7('1/Jr).(8i)f + O(N-^1 ),

where Oa ~ 8 ~0/. and Of3 ~ 8 ~ 13 • We leave the proofs of the formulas for the

rest of components to the reader as exercises.. []

Let He C M denote the hypersurfa~e {(x,y,r) E Ml r = c} (?ie is

simply a time-slice) with the metric induced by gm, where c is a constant.
By the definition of the metric gm, we have the following:
dμrt c = dμgrn ~ /\ dμgm Ol.{3

N..
Using (i -~)

2

= e-f + O(N-^1 ), we see that the volume form dμrtc of

the hypersurface is ·

dμrtc =rlf (e-foil! (1/J;dμ(M,g)) /\dμ5N+O(N-^1 )).

To find the scalar curvature of the metric on He induced by the metric
gm, first we calculate the Ricci tensor Re of the metric on He induced by
the metric 9m. Second, we pull back the tensor Re by the diffeomorphism
induced by W.

LEMMA 8.43. The Christoffel symbols f'~b corresponding to the metric

on He induced by the metric gm are given by

f'7j = r7j(M,g) mod 0 (N-^1 ),