 
Summary: Curvature formulas in §6.1 of [P]
Guofang Wei
This little note presents explicit curvature formulas in §6.1 of Perelman's paper [P]. In particular
it verifies (mod N1
) the geometric interpretation of Hamilton's matrix (trace) Harnarck quadratic
and that the Ricci tensor of the warped metric are equal to zero. The mod N1
computation of
the curvatures is also done in [STW] using Christoffel symbols. Here we do the computation using
Gauss equation and Koszul's formula.
Recall that ~M = M × SN
× R+
with the metric:
~gij = gij, ~g = g, ~g00 =
N
2
+ R, ~gi = ~gi0 = ~g0 = 0, (1)
where i, j denote coordinate indices on the M factor, , denote those on the SN
factor, and the
coordinate on R+
had index 0; gij evolves with by the backward Ricci flow (gij) = 2Rij, g is
