- PERELMAN'S FORMALISM IN POTENTIALLY INFINITE DIMENSIONS 423
(il(oi,8T)8n8j) = (R(oi,87)8n8j)
= 2 (R~~) ((o7R- 2 ~ 2 )~j+t\7iR·\7jR)+ (R?. iJ -! 2 n Vi · n VJ · R -^8 7 R-iJ ·)
1
::::: - 27 Rij + .6. 9 Rij + 2Rikj.e.Rk.e.- Rij^2 - 2,\7i\7jR^1 mod 0 ( N-1).
The components of the curvature tensor of this metric coincide (modulo
N-^1 ) with the components of Hamilton's matrix Harnack expression.5.2. Ricci flatness of (M, g). Taking the trace of the above formulas
for the curvature tensor yields.
Rc(oi, ~ 8j) = g k.e.(-R(oi, 8k)8.e., 8j ) + (-R(oi, v)v, 8j )
- tl~^13 ( R(oi, ea)e13, oj)
=
1
2 [(07R-2N2)Rc(8i,8j)+t\7iR\7jR]
2(R+fr) r- (R ~ ~) (-o7Rij - ~\7i\7jR + 2Ri.e.R.e.j)
::::: 0 mod O(N-^1 ),
Rc(oi, 87) = l.e. ( R(oi, ok)o.e., 07) + ( R(oi, v)v, 07) + tga/3 ( R(oi, ea)e13, 87)
1.
= - ( N) Rf\7jR
2 R+ 27::::: 0 mod O(N-^1 ),
Rc(on 87) = l.e. ( R(on 8k)8.e., 87) + ( R(on v)v, 87)
- ~gaf3 ( il(on ea)e13, 07)
1 l\7Rl2
4(R+fr)::::: 0 mod O(N-^1 ),
Rc(oi, ea)= gkf.( R(oi, Ok)8.e., ea)+( R(oi, v)v, ea )+~g'Y/3( R(oi, e'Y)ef3, ea)
=0,