84 3. SHORT TIME EXISTENCE
at p for any tensor Q, it follows from (3.34) that the identity
Wj (t) = gjk (t) gPq (t) ( (r g(t)):j - (I'g)~q)
= - gjk (t) gpq (t) (I'g)~q
= -tgjk (t)gpq (t) 9k£ ( (\7 9 (t))p9eq + (\7 9 (t))q9Pe - (\7 9 (t))e9pq)
= gjk (t) 9ke (8 [G (9)])e
holds at p. Thus if one defines
9-^1 : c= (T Mn)_, C^00 (T Mn)
by
one can write
W (t) = 9-^1 (8 [G (9)]).
In particular,
\liWj + VjWi = 2 (8W)ij = 2 (8 [§-^1 (8 [G (§)])])ij.
Hence the Ricci- DeTurck flow (3.33) can be written in the alternate form(3.38a)(3.38b):tg = -2 {Rc-8* [9-^1 (8 [G (9)])]},
g (0) =go.- The Ricci-DeTurck flow in relation to the harmonic map flow
It is a surprising and useful observation that equation (3.35) for the
diffeomorphisms 'Pt may be interpreted in terms of the harmonic map heat
flow.
4.1. The harmonic map heat flow. Let (Mn, g) and (Nm, h) betwo Riemannian manifolds, and let f : Mn _, Nm be a smooth map. The
derivative of f is
df = f* E C^00 (T*Mn ® j*TNm)'
where J*TNm is the pullback bundle over Nin. Using local coordinates{xi} on Mn and {y°'} on Nm, we denote the Levi-Civita connection of g
by r (g)~j and that of h by r (h)~,e· Then
df = (df)°' (dxj ® ~) = of°' (dxj ® ~).
J ay°' axJ ay°'
The induced connection\7 : C^00 (T Mn® fT Nm) --) C^00 (T Mn® T Mn® f*T Nm)
is determined by