Question: Given a regular curve $\gamma \colon I \to \mathbb{R}^n$, if we consider the variable $t \in I \subset \mathbb{R}$ as the time, then we have the usual interpretation of $\gamma'(t)$ as the (instantaneous) velocity vector at the position $\gamma(t)$ and $\lvert\gamma'(t)\rvert$ as the speed (or scalar velocity).

From this point of view, we know from differential geometry (very classical examples, indeed) that there are curves that "go through infinite length in a bounded amount of time". Of course, at these examples, the speed of these curves increase infinitely ("goes to infinity").

I have very rough ideas concerning Relativity Theory (I'm a mathematician, not a physicist) but I know that, for example, the speed a object can reach is bounded by the speed of light, and that mass, lengths and even the time are distorted at very high speed (near the speed of light).

As I said, I'm not an expert on this subject, so maybe my question even make sense, but what axioms do I insert to my models in order to get such results, from a purely mathematical/axiomatic point of view? Is the answer to this "the Einstein postulates"?

I want to know also if there is an area which study this kind of "differential geometry + relativity" (or even a "Riemannian geometry + relativity"). Would the answer to this question be simply "Relativity"? (as I said, I don't have a deep knowledge on that).

(Is there) Do you recommend any references at this subject?

Answer: 
In (special) relativity $\mathbb{R}^n$ comes with a different metric, with signature $(-,+, \ldots +)$.

The mathematical study of manifolds with a Lorentzian metric is called lorentzian geometry, or pseudo-Riemannian geometry or semi-Riemannian geometry (the latter are more general as it refers to any metric which is not definite positive).

A generic Lorentzian manifold is still not a spacetime model in General Relativity, for that you also need it to satisfy Einstein equations $G=8\pi T$.

Mathematical references are
O'Neill "Semi-Riemannian Geometry With Applications to Relativity"
Beem, Ehrlich "Global Lorentzian geometry"
Initially ignore the effects of spacetime curvature (general relativity). Consider a homogeneous (flat) real four-dimensional space. If the distance between a pair of points is given by a definite (Euclidean) quadratic form, the symmetries of space include the usual orthogonal symmetry group $\mathrm{O}(4)$ and translations (producing the [Euclidean group](https://en.wikipedia.org/wiki/Euclidean_group) $\mathrm{E}(4)$). The only difference in going to special relativity is that the quadratic form is indefinite (Lorentzian), which results in the indefinite orthogonal symmetry group $\mathrm{O}(1,3)$ and translations (producing the [Poincaré group](https://en.wikipedia.org/wiki/Poincar%C3%A9_group)). Rotations just behave a little differently, but should still be thought of as a symmetry group of four dimensional space.

Everything about parameterization of curves and velocities follows directly from this - indeed, all of the geometry of special relativity follows. A pitfall to avoid is thinking of time as universal; it is specific to the coordinate system. If you extend this local picture to differential manifolds, and you add Einstein's equation that relates the curvature of spacetime to its content (specifically the stress-energy tensor), you have general relativity.

As an aside, just as dropping the parallel postulate of Euclidean geometry produces elliptic and hyperbolic geometries, Einstein's postulates for special relativity allow for "non-flat" (de Sitter and anti-de Sitter) geometries, with symmetry groups $\mathrm{O}(1,4)$ and $\mathrm{O}(2,3)$ respectively.