Let $(M,g)$ be a spin Riemannian manifold. The coarse index of the Dirac operator $D$ lies in the $K$-theory of the Roe algebra, which I will denote by $C^*(M,g)$ since its construction uses $g$.
I understand that a quasi-isometry between $g$ and another metric $g'$ induces an isomorphism of Roe algebras that relates the coarse index of $D$ on $(M,g)$ to that on $(M,g')$.
I'm looking for some concrete examples to understand what can happen in the case when $g$ and $g'$ are not quasi-isometric.
Question 1: What are some examples of $M,g,g'$ where $C^*(M,g)$ is not isomorphic to $C^*(M,g')$?
Question 2: What are some examples of $M,g,g'$ where the coarse of index of $D$ with respect to $g$ vanishes, but that with respect to $g'$ does not?