My question, while related to the posts Ricci flow preserves isometries, The invariance of the Ricci tensor under diffeomorphisms and its non-ellipticity., and Diffeomorphism invariance of the Ricci tensor, is slightly different. Namely, when using Uhlenbeck's trick in the Ricci flow the Riemann tensor evolves by:
$\frac {\partial } {\partial t} R_{abcd} = \iota_a^i \iota_b^j \iota_c^k \iota_d^l\Delta R_{ijkl} + 2(B_{abcd} - B_{abdc} - B_{adbc} + B_{acbd})$
where $\ B_{abcd} = h^{eg}h^{fi}R_{aebf}R_{cgdi}$ and $\iota : (V,h) \rightarrow (TM,g(t)) $ is a 1-parameter family of isometries from the fixed vector space V with metric h to the evolving tangent space TM with metric g(t) such that $ h=\iota^* g(t)$ for all t.
However, according to the diffeomorphism invariance of the curvature $\iota^* Rm[g(t)]=Rm[\iota^* g(t)] $ and $ h=\iota^* g(t) $ and h is fixed. Shouldn't then
$\frac {\partial } {\partial t} R_{abcd} = \frac {\partial } {\partial t} \iota^* Rm[g(t)]=\frac {\partial } {\partial t}Rm[\iota^* g(t)]=\frac {\partial } {\partial t}Rm[h]=0 ?$
Here $\ Rm[g(t)] $ denotes the Riemann tensor of the metric g(t).
Clearly I am not understanding something correctly. What am I missing?
My reference for this material is:
Andrews, Ben, and Christopher Hopper. The Ricci flow in Riemannian geometry: a complete proof of the differentiable 1/4-pinching sphere theorem. springer, 2010.
