Why the frame will remain orthonormal under flow ?
At the beginning ($t=0$),if we choice a good frame that $g_{ij}=\delta_{ij}$,it is obvious that ${F_a}$ is orthonormal. But under the flow , $F_a(t)$ has changed, so, the pullback metric will be a constant about $t$, so,seemly there is not condition keep the frame still be orthonormal.
The below picture is from 185th page of paper.Sorry for my laze.
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The condition we want to preserve is $h_{ab} = g(F_a, F_b) = \delta_{ab}$. Since this is true at the initial time it suffices to show $\partial_t h_{ab} = 0$. The given evolution equation for $F$ is exactly what's needed for this: applying the product rule and the Ricci flow equation we get $$\partial_t h_{ab} = -2 R_{ij} F_a^i F_b^j + 2 g_{ij} g^{il}R_{lk} F^k_a F^j_b = 0.$$