1547671870-The_Ricci_Flow__Chow

4. REDUCTION TO THE ASSOCIATED ODE SYSTEM 187

Having written the evolution of the curvature operator in this form, we
can immediately apply the tensor maximum principle for systems (discussed
in Chapter 4) to get the following important result.

COROLLARY 6.27. If(Mn,g(t)) is a solution of the Ricci flow whose

curvature operator is positive (negative) initially, then that condition is pre-

served for as long as the solution exists.

4. Reduction to the associated ODE system

The maximum principle for systems introduced in Section 3 of Chapter

4 allows us to obtain qualitative information about the evolution of the

Riemann curvature operator under the PDE (6.27) by studying associated

systems (6.31) and (6.32) of ODE. This point of view originated in [59] and

will be useful below in proving pinching estimates for the Ricci curvature

of a solution to the Ricci fl.ow on a closed 3-manifold with positive Ricci

curvature.

Let (Mn, g) be a Riemannian manifold, and let { ei} be an orthonormal

moving frame defined on an open set U ~ Mn. The frame { ei} induces an

orthonormal basis { ek =et ei /\ ej} of /\^2 TU. Hence for any x EU, there

is a well defined Lie algebra isomorphism 'Px : /\^2 TxMn -+ /\^2 IRn taking the

ordered basis ( e1, ... , eN) to an ordered basis {3 = (/31, ... , f3N) of /\^2 JRn,

where N = G).

Let .R denote the space of those self-adjoint linear transformations M E

Sym^2 (/\^2 Irr) that obey the first Bianchi identity. The ODE corresponding

to the reaction-diffusion PDE (6.27) satisfied by the curvature operator of a

solution to the Ricci fl.ow may then be written as

(6.28) ~M = M^2 + M#

dt '

where M (0) = Mo E .R a nd M# is the Lie algebra square introduced in

Section 3. The basis f3 allows one to represent any M E .R by an N x N

symmetric matrix M,6 defined by

M (/3j) = {3i(M,6) ij ,

with the obvious summation in effect. The structure constants C,6 for

/\^2 TxMn ~ /\^2 IRn ~ so ( n) with respect to the basis f3 are defined by

[/3j,/3k] = f3i(C/3)ijk ·

In terms of the basis f3, the transformation M ~ Q ~ M^2 + M# is then

given by

(Q,B)ij = (M,B)ik (M,B)kj + (C,6)ipq (C,6)jrs (M,B)pr (M,B)qs.

In what follows, we shall suppress dependence on the basis {3.

Let us now specialize to the case that M^3 is a closed 3-manifold. Since

,,\.1^3 is parallelizable (as we noted in Remark 6.20) we may assume there