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Curvature formulas in 6.1 of [P] Guofang Wei
 

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 N-1
) 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 N-1
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

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics