424 8. APPLICATIONS OF THE REDUCED DISTANCERc(8T) Ba)= gk£ ( R(O-,., 8k)8e, ea)+ ( R(8r, v)v, ea)
+ ~g^111 ( R(8r, e 1 )e11, Ba)
=0,Rc(Ba, B11) = gk£ ( R(Ba, 8k)8e, e/1) + ( R(Ba, v)v, e/1)
- ~g^18 ( R(Ba, B1)e,,, e11)
- 1 (a R-2R2)
- 2 (R+~)2 r N 9a(1
= 0 mod O(N-^1 ).
Hence all of the components of the Ricci tensor are equal to zero (modulo
N-1).
Finally, taking the trace again yields the scalar curvature of g:R = 9°^0 Roo + gij Rij + -gaf1 Ra/1
1 (^2 R
2
= 2 -R8rR+l\7RI --) - (^1 N) ( D.R+-R).
2 (R + ~) T 2 R + 27 T5.3. Geometric interpretation of Perelman's entropy integrand.
Let (Mn,g (r)) be a solution to the backward Ricci fl.ow and let f satisfy
(6.15). In §6 of [297] Perelman also gave a geometric interpretation of the
integrand (6.20), i.e.,
r ( 21:::..f - IV fl
2
+ R) + f - n,of his entropy W (g, f, r). In this subsection we discuss this interpretation.
Define the diffeomorphism0 = (/; f,N : M ---+ M
by
'Pt,N '. (x, y, r) f-+ ( x, y, ( 1-~) r).
Clearly, limN--+oo 'Pt,N = idM, i.e., for N large, 'Pt,N is close to the identity.
Consider the pulled-back metric
gm~ ((/JJ,N)* g,which we think of as a perturbation of the metric g. By definition,
gm ( X, y) = g ( ( lp f,N) X, ( lp f,N) y) ·
We first compute the components of the metric -gm.