1547671870-The_Ricci_Flow__Chow

182 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

Using the usual product rule, \7 (t) and D (t) define connections on tensor
product bundles of T Mn, V, and their dual bundles T Mn and V. We
denote these connections by \7 D.
The next step in Uhlenbeck's trick is to pull back the Riemann curvature

tensor to V. For x E Mn and X, Y, Z, WE Vx, the tensor

is defined by

(6.20) (l* Rm) (X, Y, Z, W) ~Rm (l (X), l (Y), l (Z), l (W)).

Let { xk} ~=l be local coordinates defined on an open set U c Mn, and let

{ea}~=l be a basis of sections of V restricted to U. The components (l~) of
the bundle isomorphism l : (V, h) ---+ (T Mn, g ( t)) are defined by

Then the components (Rabcd) of l* Rm are defined by

How is the evolution equation for l* (Rm) related to that for Rm? In
order to answer this question, we define the Laplacian acting on tensor
bundles of T Mn and V by
n
6.D ~tr 9 ('VD^0 'VD)= L gij (\7 D)i ('VD)j,
i,j=l

where (\7 D)j (0 = 'Vj (l (~)). We then get the following formula.

LEMMA 6.22. If g (t) is a solution of the Ricci flow and l (t) is a solution

of (6.19), then the curvature l* Rm defined in (6.20) evolves by

(6.21)

f)

fJt Rabcd = fl DRabcd + 2 ( Babcd - Babdc + Bacbd - Badbc) ,

where

PROOF. Observing that