186 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE
Having completed these constructions, we can now write the evolution of
the Riemann curvature operator in a way conducive to applying the tensor
maximum principle.
THEOREM 6.26. If g (t) is a solution of the Ricci flow, the curvature
i* Rm defined in (6.20) evolves by
(6.27) gt (i* Rm)= b.DRm+Rm^2 +Rm#.
PROOF. By exploiting appropriate antisymmetries, we get, T< vpquv dpq),(rs) ( iJ) = ~ T< (gqr (sP ss - sssP) + gPS (sqsr - sr sq))
2 , vpquv i J i J i J i J
and
R pquv.LLrswx D c(pq),(rs)R (ij) rswx -- R pquv R rswx9 qr (•P>S ui Uj - ui >S>P) ujSo by (6.26), we have
(R m #) ijk£ _ - .Lvpquv.LLrswx R D C(pq),(rs)C(uv)(ij) (£k) ,(wx)
= RpquvRrswx9qr (sf Sj - Sf Sr) gvw WeSk - S£Sk)
= R~vpR~qx (sfsj - stsr) WNk - S£Sk)
= RiviRJqk - RkviRJqf, - RivjRYqk + R%vjRYq.e.·
Now recalling that Bijk£ = -R~ijR~.e.k by definition (6.15), we invoke (6.16)
in order to write Rm# as(Rm#) ijk£ = - B.e.ikj + Bki.e.j + B.e.jki - Bkj£i = 2 (Bikj£ - Bi.e.jk).
On the other hand, applying the first Bianchi identity to formula (6.23) gives(Rm^2 )ijk£ = gPqgrs ~jpsRqr£k = gPqgrs (~psj + ~sjp) (Rkrq£ + Rkq£r)
= (R;ij - R;ji) (R~k.e. - R~.e.k) = -Bij£k + Bijk£ + Bji£k - Bjik£
= 2 (Bijk£ - Bij£k).
So pulling back by the bundle isomorphism l defined in (6.19), we have(Rm^2 tbed = 2 (Babed - Babde)
and
(Rm#) abed =^2 (Baebd - Badbe) ·
Hence by Lemma (6.22), we conclude that(gt -b.D) Rabed = 2 (Babed - Babde - Badbe + Baebd)