- UHLENBECK'S TRICK 181
restrictions (lo)x : Vx ---+ TxMn are vector space isomorphisms depending
smoothly on x E Mn.) Then if we define a metric ho on V by
ho ~lo (go),we automatically obtain a bundle isometry
lo : (V, ho) ---+ (T Mn, go).
Corresponding to the evolution of the metric g (t) by the Ricci flow, we
evolve the isometry l ( t) by
(6.19a)(6.19b)a
at l =Re Ol,
l(O) =lo.Here as in (6.18), we regard the Ricci tensor Re = Re [g (t)] as a (1, 1)-
tensor. For each x E Mn, equation (6.19) represents a system of linear
ordinary differential equations. Hence a unique solution exists for t E [O, T)
(namely, for as long as the solution g (t) of the Ricci flow exists). Clearly,
l (t) remains a smooth bundle isomorphism for all t E [O, T). But more is
true.
CLAIM 6.21. D efine h (t) ~ (l (t))* (g (t)). Then the bundle maps
l(t): (V,h(t))--+ (TMn,g(t))remain isometries.PROOF. Let x E Mn and X, Y E Vx be arbitrary. Then recalling that
g , Re, and l all depend on time, we compute that:th (X, Y) = :t [(l*g) (X, Y)]
a= at [g (l(X), l (Y))]
= -2Rc(l(X),l(Y))
+ g (Re (l (X)), l (Y)) + g (l (X), Re (l (Y)))= 0.
The Levi-Civita connections
\1 (t) : C^00 (T Mn) x C^00 (T Mn)---+ C^00 (T Mn)of the metrics g ( t) pull back to connections D ( t) on V defined by
D (t) ~ l (t)* \1 (t) : C^00 (T Mn) x C^00 (V) ---+ C^00 (V),where for XE C^00 (T Mn) and~ E C^00 (V), we have
(l*\l) (X, ~) ~ (l*V')x (~) ~ V' x (l (~)).D