Apologies if I get the notation wrong. Still learning this stuff.
Suppose I have a 2 dimensional Riemannian manifold $\mathcal{M}$ that is covered by a single chart: $\phi: \mathcal{M} \rightarrow \mathbb{R}^2$ (or alternatively, let's just think about a single chart).
I can think of this manifold as just the plane that is stretched out in some places and/or shrunk in other places (or directions) depending on the metric $g$. I understand enough calculus and physics to know what to do when someone hands me some metric that depends upon coordinates, say $g = g_{xx}(x,y) dx^2+g_{xy}(x,y) dx dy+ g_{yy}(x,y) dy^2$ So that in that chart, I can do calculations (find the length, etc). But I still struggle with intuition. So here's my question:
Suppose I have some rectangle in the plane with some such metric prescribed. Can I think of this (for the 2D case) as just just like a topographic map? If not, how is the changing metric different from the bumps and valleys of such a map? Any example to help me understand this better will be very much appreciated.
You have to define what a "topographic map" means. Once you do, I think you will end up with the requirement that your surface is isometric to the graph of a function in 3-space with Riemannian metric induced from the flat metric on $R^3$. This is fine for some intuition but not enough to recover all Riemannian metrics. For instance, hyperbolic plane does not embed isometrically in the flat 3-space.
Edit: Furthermore, there are smooth Riemannian metrics which do not admit isometric (smooth) embeddings in $R^3$ even locally, see, say, here.