Geometry of Metric Spaces I'm reading a book on Metric Spaces and the author is always talking about the "geometry" of some metric spaces, but he doesn't say what he means by geometry.
For example:

Despite the fact that it is infinite-dimensional, the next example shares many nice geometric properties with the real line $\mathbb{R}$.
$\textbf{Example}$: $(l_\infty)$ This is the space whose elements consist of all bounded sequences $(x_1,x_2,..)$ of real numbers, with the distance $d_\infty (x,y)$ between two such sequences $x=(x_1,x_2,...)$ and $y=(x_1,x_2,...)$ taken as $d_\infty (x,y)=\sup\limits_{1\leq i <\infty}|x_i-y_i|$.

I think he is talking about some theorems that are true on both metric spaces and he considers to be of a "geometric flavor", but he gives no example of such theorems.
What does the author mean by "geometry" in this case?
If I'm right, examples of such theorems would be nice.

If any two points can be linked by a path way in a metric space, we can stablish the notion of path lenght and geodesic paths. If you require more, you can also define angles between two paths. Then you can stablish a kind of geometry in this metric spaces. Metric spaces like those are called Geodesic Metric Spaces.
