418 8. APPLICATIONS OF THE REDUCED DISTANCE
Recall from (7.11) that given a solution (Mn,g(r)), TE (O,T_h of
the backward Ricci flow, Perelman [297] introduced the manifold M
M x SN x (0, T) with the following metric:
9ij = 9ij, 9af3 = T9af3, - 900 = N
27- R, 9ia = 9io = 9ao = 0,
i.e.,
g = 9ijdxidxj + T9af3dyadyf3 + ( R + ~) dr^2 ,
where the metric 9af3 on sN has constant sectional curvature 2 1N.
5.1. Riemann curvature tensor of (M, g). We shall apply the fol-
lowing two steps to compute the Riemann curvature tensor of the mani-
fold (M,g). First we treat (M,g) as a hypersurface in the manifold M =
M x (0, T) with the metric
g = 9ijdxidxj + ( R+ ~) dr^2 •
We compute the curvature of the manifold (M, g) using the Gauss equa-
tions and Koszul's formula. Secondly, we consider the manifold (M, g) as a
warped product with base (M, g) and fiber SN. We then use O'Neill's formu-
las to compute the curvature of g. This method of computation essentially
follows Guofang Wei [368].
Let ai ~ 8 ~i denote the coordinate vector fields on the M factor, let
aT ~ gT, and let
v = i a
(R + :fr)l/2 Tbe the unit normal vector field of M x { r} C M. Direct computation, using
[8n ai] = 0, gives
\liR
[v, 8i] = ( N) 1/.
2 R+ 2T(8.57)By the formula for the evolution of the metric of a hypersurface evolving in
the direction of its normal with speed J 87 J = ( R + fr)^112 , we have^8
2~j = aT9ij = 21aT1 · nij,
where II denotes the second fundamental form of M x { r} C M.^9 Therefore
1II(ai,aj) = IIij = N 1/2Rij,
(R + 2T)
and the Levi-Civita connections of g and g are related by
(^8) See the proof of (B.13) with the mean curvature H replaced by 187 1.
(^9) In other parts of this volume we have sometimes used h instead of II to denote the
second fundamental form.