76 3. SHORT TIME EXISTENCE
Consider the composition
D (Reg) o o; : C^00 (T Mn) -+ C^00 (S 2 T Mn).
This is a priori a third-order differential operator, so its principal symbol
is the degree 3 part of its total symbol. But as we observed in Section 2.2,
the invariance (3.11) of the Ricci tensor under diffeomorphism implies that
[ (D (Reg) o o;) (X)Lk = ~ [.Cx" (Re g)]jk.
Since the right-hand-side involves only one derivative of X, we see that its
total symbol is at most of degree 1 in (. In other words, the principal (degree
3) symbol 0-[(D (Reg) o o;)J (()is in fact the zero map. Since by (3.9), one
has
o =a-[(D (Reg) o o;)J (() = O-[D (Reg)](() o a-[o;J ((),
it follows that
im ( 0-[ o; J ( ()) s;;; ker ( 0-[ D (Re g)] ( ()) ,hence that 0-[ D (Reg)] ( () has at least an n-dimensional kernel in each
(n(n + 1)/2)-dimensional fiber:
(3.23) dim (ker (0-[D (Reg)](())) ::=: n.REMARK 3.12. In Section 2.2, we proved that the diffeomorphism in-
variance of the Ricci tensor implies the contracted second Bianchi identity.
They are actually equivalent, since commuting derivatives and applying the
contracted second Bianchi identity reveals that
[(D (Reg) 0 o;) (X)] jk = lvpvj (V'kXp + Y'pXk)
1
+ 4VP\7k (V'jXp + Y'pXj)
1 1- 4~ (Y'jXk + V'kXj) + 2V'jV'k (ogX)
1
= 4XP (V'jRkp + V'kRjp + V'qRqjkp + V'qRqkjp)
- ~ ( R~V'kXp + R~Y'jXp)
_1( - 2 X p Y'pRjk + Rj p V'kXp + Rk p Y'jXp )
1
= 2 [.Cx~ (Re g)]jk.Now consider the linear operator