- 1-AND 2-LOOP VARIATION FORMULAS RELATED TO RG FLOW 33
where the subscripts 1, 4 denote the components on which the operator is
acting. Note that if Rc+\7\7f:::::::: 0, then H\lf:::::::: 0.
Recall that under(17.99a)(17.99b)a
atg = -2Rc,
af 2
at = -flf-R+ l\7fl ,
we have (see Proposition 6.95 in Part I)
(17.100)(gt +flL - 2\7f · \7) (Rc+\7\7!) = 2Hvf - (Rc+\7\7f) © (Rc-\7\7!),
where, for (1, 1)-tensors A{ and Bf, we define (A© B){ ~A~ BL+ Bf A{.
Define
P* (X, Y, Z) ~ P (Z, Y,X),so that (see (15.50) in Part II)
div (Rm)= P*,
where div (Rm) = trace~'^2 (\7 Rm) and where the superscripts 1, 2 indicate
that the first two components are traced.
Under the Ricci flow, the Riemann curvature (3, 1)-tensor Rm evolves
by (see (6.2) in Volume One)
(17.101) :t Rm= - [\7, ~] Rc+dvP*,
where (dvP) (X, Y, Z, W) = (\7xP) (Y, Z, W) - (\7yP*) (X, Z, W).
Define the symmetric 2-tensor
(17.102) a~ "l: Rm(·, ei, ej, ek) Rm(·, ei, ej, ek),
i,j,kwhere { ee} is an orthonormal frame, i.e., CTij = Rik£pRjk£p· Then a satisfies
the contracted second Bianchi-type identity
(17.103) div (a - l 1Rml^2 g) =Rm(·, ei, ej, ek) P* (ei, ej, ek).
Now we discuss some calculations due to two of the authors [44]. By
(17.101), under the Ricci flow
1 a 24 at IRml = (Re, a) + (Rm, \7 P*)
(17.104) = (Re, a)+ div (Rm(·, ei, ej, ek) P* (ei, ej, ek)) - IPl^2.
We then compute that under (17.99)
(17.105) ~D (1Rml^2 e-f) = e-f (-L (a, \7 f) +IP - iv1 Rml^2 ),