448 A. BASIC RICCI FLOW THEORY
LEMMA A.1 (Metric variation formulas). Suppose that g (s) is a smooth
1-parameter family of metrics on a manifold Mn such that gsg = v.
(1) The Levi-Civita connection r of g evolves by(Vl-3.3)(Vl-3.5)(Vl-p. 69b)a 1
-a Rjk = -gpq (\Jq\ljvkp + \lq\lkvjp - \lq\lpvjk - \lj\lkvqp)
s 2= -t [.6.LVjk + \lj\lk (trgv) + \lj (8v)k + \lk (8v)j],
where .6.L denotes the Lichnerowicz Laplacian of a (2, 0)-tensor,
which is defined by
(Vl-3.6) (.6.Lv)jk ~ .6.vjk + 2gqp R~jk Vrp - gqp Rjp Vqk - gqp Rkp Vjq·
(4) The scalar curvature R of g evolves by(Vl-p. 69c)(Vl-p. 69d)as a R = lJ ·· g ke (-\li\ljVkf, +\Ji \lkVjf, - VikRje)
= -.6.V +div (divv) - (v, Re),
where V ~ gijVij is the trace of v.
(5) The volume element dμ evolves by
a v
(Vl-p. 70) as dμ = 2 dμ.(6) Let "(s be a smooth family of curves with fixed endpoints in Mn and
let Ls denote the length with respect tog (s). Then
(Vl-3.8)! Ls ('Ys) = t 1 v (T, T) dO" -1 (\lrT, U) dO",
?s ?sw h ere O" is. arc l eng th , T ..!... "7" O?s ea, an d U ..!... "7" as^0.
1.6. Commuting covariant derivatives. In deriving how geometric
quantities evolve when the metric evolves by Ricci flow, commutators of
covariant derivatives often enter the calculations. (For the following, see
p. 286 in Section 6 of Appendix A in Volume One.) If X is a vector field,
then
(Vl-p. 286a)