Hamilton-DeTurck Ricci flow : $\partial_{\lambda} g_{\mu \nu}=-2 R_{\mu \nu}+\nabla_{\mu} V_{\nu}+\nabla_{\nu} V_{\mu}$. i now have studied some paper , i think the main reason is that the original Ricci flow is not invariant under coordinate transformation. Here is the one explanation from a paper, which shows that the Hamilton-DeTurck Ricci flow will generate a vector (V) along flow,which ensure that flow equation keeps invirant under any coordinate transformation . I wonder if it is the main reason why give rise up to Hamilton-DeTurck Ricci flow ?
The Ricci flow equation IS invariant under coordinate transformation.
I think the main motivation is from PDE. Since the RF is degenerate parabolic, the existence and uniqueness are not immediate from "standard" PDE theory, even when restricted on compact manifolds.
This is why Hamilton made a huge effort (The Hard Implicit Function Theorem) in proving the existence of RF in here. He explained the degeneracy starting at p.260.
Shortly after this paper, Deturck gives another method to prove the existence. In particular, the Detruck Ricci Flow is introduced. The Deturck Ricci Flow is strictly parabolic and differs from the Ricci Flow only by a family of diffeomorphisms. Note that his proof is essentially just 3 pages long.
The DeTurck Ricci Flow is essential in the non-compact situation. It is used both in proving the existence and uniqueness.
The DeTurck trick is also generalized to other geometric flows like the Mean Curvature Flow.