- IRREDUCIBLE COMPONENTS OF 'i7^3 v 109
Since IAl
2= ~ IV'vl
2
and/3 \ 23 3 2 2
\ V' v, A 1 = - 3 ('Vijkv - 'Vjikv)'Vjv(gs2 )ik = -
3
IV'v l ,we conclude that(29.130)Let TF (/3) stand for the totally trace-free part of a covariant 3-tensor /3. We
have the orthogonal irreducible decomposition(29.131a)(29.131b)where trp,q denotes the t race with resp ect to gs2 acting on a tensor by metrically
contracting its p-th and q-th components.
Note that from (TF(B), Z ® gs2) = 0 we see that(29.132) IBl^2 = ITF(B)l^2 + 121z1^2.
Substituting (29.130) into this, we obt ainLEMMA 29.40. The trace-free part of B = S(V'^3 v) satisfies(29.133) ITF(B)I^2 = I V'^3 v 12 -^3 2 2
4
IV' 6.s2vl - IV'v l - (V' 6.s2v, V'v).Define the key quantity(29.134) Q ~ v ITF(B)l^2.Note that Q = 0 on a round 2-sphere. In t he next section we shall compute the
evolution of Q under the Ricci fl.ow on 52.
EXERCISE 29.41. Let Pij ~ 'V;iv + v(gs2 )ij and P ~ gii Pij. Show that
1
TF(B)ijk = 'ViPik - 4 ('ViP(gs2)ik + 'ViP(gs2)ik + 'V kP(gs2)ii).Show also that 'ViPjk is totally symmetric. Hence
TF(B) = S ( V'(p - ~Pg)).REMARK 29.42. Alternatively, we can obtain TF(B) as follows. The irreducible
decomposition for V'^3 v on 52 is given by
(29.135)where the 1-forms X and Y are defined so that
0 = tr^2 •^3 (H) = V' 6.s2v + 2X + 2Y and 0 = tr^1 •^2 (H) = 6.s 2 'V v + X + 3Y.
This implies that
(29.136) andV' 6.s2v dv
X=- +-.
4 2