I'm trying to understand basic differential geometry (my background is in mathematical logic), and I'm having a bit of difficulty with a basic point. Frequently we want to consider the set of metrics on a given manifold modulo some equivalence relation (generally, isometry), but I can't visualize metrics very easily. Is there some example of a reasonably simple manifold, and two "very different" metrics on it, that would help me understand what spaces of possible metrics can look like?
Thanks in advance