1547671870-The_Ricci_Flow__Chow

188 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

exists a globally defined orthonormal moving frame {ei}· We fix an or-

thonormal basis { ek = Bfj ei /\ ej} of /\^2 T M^3. For example, such a basis is

given by taking

where * : /\^1 T M^3 ---) /\^2 T M^3 corresponds to the Hodge star operator. In

dimension n = 3, the Lie algebra square is easily computed. Indeed, it is

readily verified that ([ei,ej] ,ek) is fully alternating in (i,j,k), hence that

(6.29)

(

a b e ) # ( df - e

2

ee - bf be - ed )

b d e = ce - bf af - e^2 be - ae.

e e f be - ed be - ae ad - b^2

Now we identify Rm with the quadratic form M defined on /\^2 T M^3 by

M (ei /\ ej, ee /\ ek) = (R (ei, ej) ek, ee).

Using the basis { B^1 , B^2 , e^3 } of /\^2 TU, we further identify M with the matrix
(Mpq) defined on each fiber /\^2 TxM^3 of the bundle /\^2 T M^3 by

(6.30)

If { ei} evolves to remain orthonormal, the PDE (6.27) governing the behavior
of Rm corresponds to the ODE

(6.31)

satisfied by M in each fiber. If { ei} is chosen so that Mo is diagonal at
x E M^3 with eigenvectors >. (0) 2:: μ (0) 2:: v (0), then (6.29) and (6.31)
combine to yield the system

(6.32)

posed on JR.^3. In particular, M (t) remains diagonal (which is not in general

the case in higher dimensions). Moreover, >. ( t) 2:: μ ( t) 2:: v ( t) for all t 2:: 0