TITLE: Geometry of Metric Spaces
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: I think the example given by the author is a bad example to testify for his thesis. Here is a better one: The geometries on the euclidean plane and on the surface of a sphere are certainly different, but they share many interesting properties. For instance, given three (sufficiently small) lengths $a$, $b$, $c$ satisfying the triangle inequality there is, up to isometry, exactly one triangle having $a$, $b$, $c$ as side lengths.