Ricci flow preserves holonomy Could someone please give me a reference where I can find a complete proof of the result Ricci flow preserves holonomy? Is there any way to prove that Ricci flow preserves Kahler condition without using that theorem?
 A: Actually, the only nontrivial case is the Kahler case:  
If you put on the assumptions you need in order for the Ricci flow to be uniquely defined, then by passing to a cover, you can assume that $(M,g)$ is the direct product of $(M_i,g_i)$, each factor having irreducibly acting holonomy (de Rham Splitting Theorem).  Thus, you are reduced to the case that the holonomy acts irreducibly.
Assuming that $(M,g)$ has irreducible holonomy, then, by the Berger Holonomy Classification Theorem, there are only three possibilities:  (i) the holonomy of $(M^n,g)$ is $\mathrm{SO}(n)$, (ii) the holonomy of $(M^{2n},g)$ is $\mathrm{U}(n)\subset\mathrm{SO}(2n)$, or (iii) the metric $g$ is Einstein (or even Ricci-flat).  In cases (i) and (iii), there is nothing to prove.  The case (ii) is the Kähler case, and, as has been pointed out in the comments, you just need to check that the Ricci-flow in the Kähler case can be reformulated as a flow on the corresponding (1,1)-form, i.e., the underlying complex structure does not change.  Then uniqueness does the rest.
