1547671870-The_Ricci_Flow__Chow

3. THE STRUCTURE OF CURVATURE EVOLUTION 183

we compute

3. The structure of the curvature evolution equation

By looking more closely at the structure of the Riemann curvature op-

erator, we can obtain valuable insight into the structure of its evolution

equation (6.21).

We begin with the observation that although B is quadratic in Rm, it is

not the square of Rm. A natural way of defining the square is first to regard

Rm as the operator on 2-forms

Rm : /\^2 T* Mn --+ /\^2 T* Mn

defined for all U E /\^2 T* Mn by

(Rm(U))ij ~ -gkp/q~jkl!Upq·

The operator Rm is then self adjoint with respect to the inner product

defined for all U, V E /\^2 T* Mn by

(6.22) (U, V) ~ gikgjl!Uij Vke;

indeed, it follows easily from symmetries of the Riemann curvature tensor

that

(Rm (U), V) = (U, Rm (V)).

Note that

Rm = 2"' id /\2T* Mn

in case (Mn, g) has constant sectional curvature "'' because then

Rijkf! = "'(9il!9jk - 9ik9jf!) ,