- THE EVOLUTION OF CURVATURE 177
This identity shows in particular that the Ricci tensor determines the Rie-
mann tensor when n = 3. Because of this, the reaction terms in ( 6. 7) depend
only on the Ricci tensor.
LEMMA 6.10. In dimension n = 3, the Ricci tensor of a solution to the
Ricci fiow evolves by
(6.10) ot a Rjk = b.Rjk + 3RRjk - 6gpq RjpRqk + ( 2 IRcl^2 - R 2) gjk.
PROOF. Substituting (6.9) into the second term on the right-hand side
of ( 6. 7) yields
2gpq grs RpjkrRqs = ( 2 IRcl^2 - R^2 ) gjk - 4gpq RjpRqk + 3RRjk·
0
The weak maximum principle for tensors (discussed in Chapter 4) im-
plies the following important consequence of formula (6.10).
COROLLARY 6.11. Let g (t) be a solution of the Ricci fiow with on a
3 -manifold with g (0) = go. If go has positive (nonnegative) Ricci curva-
ture, then g (t) has positive (nonnegative) Ricci curvature for as long as the
solution exists.
PROOF. At any point and time where the Ricci tensor has a null eigen-
vector, it has at most two nonzero eigenvalues. Then one has 1Rcl^2 2': R^2 /2,
whence the result follows immediately from Theorem 4.6. 0
REMARK 6.12. More generally, in any dimension n 2': 3, the decomposi-
tion (6.8) lets one write the evolution equation (6.7) satisfied by the Ricci
tensor in the form
a 2n e 2n
otRjk = b.Rjk - n _ 2 RjRek + (n _ l) (n _ 2 ) RRjk- n: 2 (1Rcl^2 - n ~ 1 R^2 ) gjk + 2RPqWjpqk·
Since 1Rcl^2 2': R^2 / (n - 1) whenever Re has a zero eigenvalue, one concludes
in particular that the Weyl tensor is the sole obstacle to preserving the
condition Re 2': 0.
1.3. The Riemann curvature tensor. The observations made in the
previous two subsections suggest that the Riemann curvature tensor also
satisfies a heat-type (reaction-diffusion) equation.
LEMMA 6.13. Under the Riccifiow, the (3, 1)-Riemann curvature tensor
evolves by(6.lla)(6.llb):t Rfjk = b.Rfjk + gpq ( RijpR;qk - 2R;ikRJqr + 2R~irRJqk)
- Rf R~jk - RjRfpk - R~Rfjp + R~Rfjk·