Riemannian metrics and how spaces look I thought I had a fairly good understanding of Riemannian metrics until I came across this exercise in Petersen's book.

Construct paper models of the Riemannian manifolds ($\mathbb{R}^2, dt^2 + a^2t^2d \theta ^2$). If $\alpha = 1$, this is of course the Euclidean plane, and when $\alpha < 1$, they look like cones. What do they look like when $\alpha >1$?

I fail to understand why changing the numbers that you assign to a pair of vectors in a tangent space (changing the Riemannian metric) would make a figure look different. If I give $\mathbb{R}^2$ Cartesian coordinates why would changing the inner product cause any difference in the way it looks? Is there some intuition behind Riemannian metrics that I am missing? Thank you.
 A: The factor $a$ makes the length of each circle centered at the origin equal to $2\pi a r$, where $r$ is the radius. 
Case $a<1$: not enough circumference
To realize this case in practice, cut off a part of the circle, specifically $(1-a)$ part of it (in angle terms, $2\pi (1-a)$).  This is well explained in wikiHow: 

Note that  this  surface is inclined to close in on itself; this is a manifestation of positive curvature (concentrated in the vertex here).
Case $a>1$: too much circumference.
Instead of removing a part of a circle, we need to add more of it. A way to achieve this is to take $2$, $3$, or more sectors of the kind shown above, and glue their edges so that a single surface is obtained.  With $n$ sectors of angular size $\theta<2\pi$, you get $n\theta$ total angle, which can be any positive number you want. 
The sectors (labeled $1,2,\dots,n$ and put in a stack)   should be glued so that, say, the bottom end of 1st is glued to the top end of 2nd, the bottom end of 2nd to the top end of 3rd... and the bottom end of $n$ to the top end of $1$st. This last gluing step cannot actually be done in a three-dimensional space without creating self-intersection. You'll just have to imagine that the self-intersection isn't there: the surface gets around another part of itself via a detour into the 4th dimension. Here is an illustration from Wikipedia.

Unlike the first example, the surface wants to spread around (so much that there isn't enough room for it in 3D space); this is a manifestation of negative curvature. 
