484 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS
evolution of the metric is given by
:t9ij = ( D ~; ( aa~) , ~~) + ( ~~, D ~~ ( ~~) )
= -H ID ax v, ax) -H I ax' Dax v)
\ axi axJ \ axi axj
= -2Hhij,where we used (v, g;;) = ( g;, v) = 0. This is (B.13).
Since ( ~ , v) = 0 and.
o =gt (v, ~;) = ( D8aYf v, ~~) + \v, Dg; (-Hv))
= / \ Daxv, at ~X:)-uxi ~~' uxi
the normal v evolves by
.. aHaX
(B.19) Daxv at = lJ~~ uxi uxJ = \!H,where \1 His the gradient on the hypersurface of H. This is (B.14).
The evolution of the second fundamental form is (we use [^88 1, g;;] = 0)
~ hij = - ~ / DQK. ~X:, v)
ui ui \ ax' uxJ= - /Dax (Dax ~X:), v)- /Dax ~X:, Daxv)
\ at ax' uxJ \ axi uxJ at.= - ( D g; (D g~ ( aa~)) 'v) -(Rm P ( aa~' ~~) ~~' v)
- /Dax ~X,Daxv)
\ axi uxJ at= / DM (~H_v+HDn_v) ,v) +H/Rmp (v, ~X:) ~X:,v)
\ ax• uxJ axJ \ uxi uxJ- /Dax ~X:, !H)
\ ax• uxJ
a^2 H ke ( ax ·) · k aH
=a xi ·a. xJ +Hhjk9 Dax~ ax• uX" n)V +H(Rmp)ViJ'v-riJ'~ ux k
= \liVjH - Hhjklehei + H (Rmp)vijv,
where we used (B.11), (B.19), and (B.12), and whererij k = g ke ( Dax~'~ ax ax) R
ax• uxJ uX
are the Christoffel symbols and \1 is the covariant derivative with respect
to the induced metric on the hypersurface. This is (B.15).