1547671870-The_Ricci_Flow__Chow

180 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

2. Uhlenbeck's trick

Let (Mn,g(t)) be a solution of the Ricci fl.ow, and let {e~}:=l be a

local frame field defined in an open set U C Mn. Suppose that { e~} is
orthonormal with respect to the initial metric go. We would like to evolve

this frame field so that it stays orthonormal with respect tog (t). This may

easily be done by considering the following ODE system defined in TxMn for
each x EU:

(6.18a)

(6.18b)

d

dt ea (x, t) =Re (ea (x, t)),

ea (x,O) = e~ (x).

Here we regard the Ricci tensor Re = Re [g (t)] as a (1, 1)-tensor, that is,

an endomorphism Re : TMn -> TMn. Since (6.18) is a linear system of

n ordinary differential equations, a unique solution exists as long as the

solution g (t) of the Ricci fl.ow exists.

LEMMA 6.17. The inner product g (ea, eb) is constant for any solution

g ( t) of the Ricci flow and any frame field {ea ( t)} satisfying { 6.18).

PROOF. Noting that g, Re, and each ea all depend on time, we compute

that

gt [g (ea, eb)] = ( gtg) (ea, eb) + g (gt ea, eb) + g (ea, gt eb)

= -2 Re (ea, eb) + g (Re (ea), eb) + g (ea, Re (eb))

=0.

D

COROLLARY 6.18. If { e~} is orthonormal with respect to go, then {ea (t)}

remains orthonormal with respect tog (t).

REMARK 6.19. If Mn is parallelizable (in other words, if T Mn is a trivial

bundle) there exists a global frame field that is orthonormal with respect to

any given Riemannian metric g. Indeed, one may take any global frame field"

and apply the Gram- Schmidt process (which preserves smoothness of the

frame field).

REMARK 6.20. Every closed 3-manifold is parallelizable, hence may be

assigned a global orthonormal frame field.

The idea of evolving a frame field to compensate for the evolution of a

Riemannian metric has a more abstract formulation, due to Karen Uhlen-

beck, which we now present. The Uhlenbeck trick allows us to put the

evolution equation (6.17) satisfied by Rm into a particularly nice form. We'

begin as follows. Let (Mn,g(t): t E [O,T)) be a solution to the Ricci fl.ow

with g (0) =go. Let V be a vector bundle over Mn isomorphic to TMn,

and let io : V -> T Mn be a bundle isomorphism. (In other words, the