10 1. THE RICCI FLOW OF SPECIAL GEOMETRIES
If we want to evolve g by the Ricci flow, we must study its curvature. For
brevity, we display the map ad* in the form
(ad F1 ) F1 (ad F2 ) Fi (ad F3 )* F1
(ad Fi)* F2 (ad F2)* F2 (ad F3)* F2(adF1)* F3 (adF2)* F3 (adF3)* F3If {Fi} is a Milnor frame, it is easy to compute that ad* is determined by
0 2>.~F3 -2>.~F2(1.3) -2μfbF 3 02v~F2 -2v~F1 0
The Levi-Civita connection of g is determined by ad and ad* via the formula(1.4) 'VxY = ~ {[X, Y] - (adX) Y - (ad Y) X}.
Then once the connection is known, it is straightforward to compute that
the curvature tensor of g is given by(R (X, Y) Y, X) = ~ i(adX)' Y +(ad Y)' Xl
2
- ((adX)' X, (ad Y)' Y)
3 2 1 1 - 4 l[X, Y]I - 2 ([[X, Y], Y] ,X) -
2
([[Y,X] ,X], Y).LEMMA 1.16. Suppose that {Fi} is a Milnor frame for a left-invariant
metric on a 3-dimensional unimodular Lie group g. Thenfor all k and any ii-j.
PROOF. Without loss of generality, we may assume that i = 1, j = 2,
and k = 3. Using ( 1.1), ( l. 3), and ( 1.4), and noting in particular that
\7 Fe Fe= - (ad Fe)* Fe= 0 for any R, we compute that(R(F3,F1)F2,F3) ~ ('VF 3 (\7FiF2)-'VF 1 ('VF 3 F2)-'V[F 3 ,FiJF2, F3)
= ('VF 3 F2, 'VFiF3) - ('VF 1 F2, 'VF 3 F3)- ('V[F 3 ,Fi]F2, F3)
= ('VF 3 F2, 'VF 1 F3)
= ~ ( ( ->. - μ! + v ~) F1, ( -μ - v ~ + >. ~) F2)
= o.D