1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

460 A. BASIC RICCI FLOW THEORY

LEMMA A.23. If we evolve the isometry l (t) by

(Vl-6.19a)

(Vl-6.19b)

then the bundle maps

remain isometries.^4

a

-1,=Rco1,,

at

1, (0) =lo,

1,(t): (V,h)---+ (TM,g(t))

We define the Laplacian acting on tensor bundles of TM and V by

b.n ~ tr 9 (\Jn o \Jn),

where (V n) x ( ~) = /, -^1 (V x ( /_, ( ~))) acting on sections of V and \Jn is nat-
urally extended to act on tensor bundles. For x E M and X, Y, Z, W E Vx,
the tensor

(Vl-p. 182)

is defined by

(Vl-6.20) (i* Rm) (X, Y, Z, W) ~Rm (i (X), i (Y), i (Z), i (W)).

Let Rabcd denote the components of 1_,* Rm with respect to an orthonormal

basis of sections of V We have

8

(Vl-6.21) at Rabcd = b.nRabcd + 2 (Babcd - Babdc + Bacbd - Badbc),

where

B abed _,_ -;--heghfhR aebf R cgdh·

We may rewrite the above equation in a more elegant way. (See pp.

183-187 in Section 3 of Chapter 6 in Volume One.) Let fl be a Lie algebra

endowed with an inner product (-, ·). Choose a basis {cp°'} of fl and let c:;f3

denote the structure constants defined by [ cp°', cpf3] ~ 2..::: 7 c:;f3 cp^7. We define

the Lie algebra square L# E fl @s fl of L by

(Vl-6.24) (L#)af3 ~ ct^5 C~(, L 7 E:Lo<,·

For each x E Mn, the vector space A^2 T; M cari be given the structure of a

Lie algebra fl isomorphic to so (n). Given U, V E A^2 T;M, we define their

Lie bracket by

(Vl-6.25)

THEOREM A.24 (Rm evolution after Uhlenbeck's trick). If g (t) is a

solution of the Ricci flow, then the curvature 1_,* Rm defined in (Vl-6.20)

evolves by

(Vl-6.27)

(^4) The statement here is the one ~e intended in Claim 6.21 of Volume One.