Suppose $(M,g)$ is a closed Riemannian manifold and $f$ is a Morse function whose critical point are isolated points (not Morse-Bott). If the Morse-Smale condition is satisfied, then we can define a Morse (chain) complex $C_*(M,f,g)$ out of these data. If we have two different metrics $g_1,g_2$ both satisfying the Morse-Smale condition with $f$, then it is well known that we can define chain maps from $C_*(M,f,g_1)$ to $C_*(M,f,g_2)$ all of which induce a canonical isomorphism on homologies.
I am curious about what happens on the chain complex level. The generators are the same but the differentials for $C_*(M,f,g_1)$ and $C_*(M,f,g_2)$ might be different.
Are there any papers discussing on this? Are there any examples where the complexes are really different or examples where, if we put more restrictions on $f,g_1,g_2$, then the complexes must be the same?
