I know some PDE theory for nice open domains in $\mathbb{R}^n.$ I want to know what the changes are when I switch to other domains like manifolds.

For example, do things like Poincare's inequality for $H^1_0$ hold, or like the norms $|\Delta \cdot |_{L^2}$ being equivalent to $|\cdot|_{H^2}$ is still true?

My friend told me that nothing changes since on manifolds we just work locally in $\mathbb{R}^n$ but I think he is not entirely correct.

Also, I want to know whether the PDE theory I know (eg. Evans) I can apply it to surfaces on $\mathbb{R}^n$ like a sphere. Or if not a sphere some open surface. Or do I need to use the PDE on manifold theory.

Thanks for any replies