Are Isometric Metric Spaces Identical?

[Whovian]

While trying to come up with a way to disprove [censored conspiracy theory that I think about for amusement] for the billion-and-first time, something occurred to me. After a bunch of bantering with myself over how exactly it works, the problem came down to this. I'm somewhat of an idiot when it comes to differential geometry and haven't studied it in any sort of rigour, but here goes.

If we have two same-dimensional spacetimes with not necessarily identical distortions (or whatever the proper term is,) where distance along a geodesic is defined, if we can find a bijection $f$ from the set of geodesics in one spacetime to the set of geodesics in the other and another bijection $g$ from the points in one spacetime to the point in the other such that the distance along a geodesic $l$ between points $P$ and $Q$ is the same as the distance along $f\left(l\right)$ between $g\left(P\right)$ and $g\left(Q\right)$, are the two spacetimes, in a sense, "identical," however that's defined? Are there any extra "this behaves nicely" conditions needed, such as continuity of $f$ and $g$? Or am I just nuts?

EDIT: Oh, and apologies if this is the wrong forum, I just took my best guess as to which forum to put this in. If this is the wrong forum, can it please be moved?

EDIT 2: Sorry if I sounded like a confusing idiot, I sort of am when it comes to stating things like this.

 

[bossman27]

You're pretty much correct. The two spacetimes are metric spaces, in the sense that they are sets for which a distance between any two points is defined by some metric. There are several notions of equivalence between two metric spaces:

If there exists a continuous bijection between the two spaces, in both directions, we say they are "homeomorphic" (or topologically isomorphic). This just means there is a one-to-one mapping between points in the spaces.

Given that a bijection and its inverse (i.e. both directions) are uniformly continuous, the spaces are called "uniformic." Any isometry (distance-preserving map) between metric spaces is uniformly continuous by definition.

If there exists a bijective isometry between the two spaces (say, $X$ and $Y$ with metrics $d_{X}$ and $d_{Y}$), they are called "isometric." This just means that the bijective function/map $g: X \rightarrow Y$ is distance preserving, such that for any $p, q \in X$:

$d_{Y}(g(p), g(q)) = d_{X}(p, q)$

...and as you guessed, this last property means that the two spaces are essentially identical.