Non-Euclidean geometries are generally presented as models for the first four of Euclid's axioms, which do not satisfy the fifth axiom (parallel's postulate).
Most texts present two models of non-Euclidean geometries: hyperbolic geometry and spherical geometry.
These two models can be considered as instance of Riemannian manifolds of dimension 2 (surfaces), that are special in the sense that they have a constant curvature (negative in the case of hyperbolic geometry, positive in spherical geometry).
My question is: why do we only consider Riemannian manifolds of constant curvature. Why couldn't we have a model of non-Euclidean geometry which would be a surface of non-constant curvature?
Best Answer
Despite its name, "non-Euclidean geometry" is very much tied up to how Euclid did things. Its starting point is to discard the fifth postulate specifically, but keep everything else like Euclid would have done it, and then see what happens.
In particular, one thing Euclid did -- not often, but in a few fundamental places such as the proof of Proposition I.4, the SAS theorem -- is to say something like "imagine we move one triangle over so its one vertex covers the vertex in the other triangle". He has no explicit definitions or postulates that promise us this is something that makes sense, but he does so anyway. Depending on your temperament, we can consider this at best a hidden assumption, or at worst a proof by handwaving.
(See e.g. the commentary at http://aleph0.clarku.edu/~djoyce/elements/bookI/propI4.html)
This tacit assumption that we can move a geometric shape to any chosen place and orientation without stretching or tearing it, is also made in the classical development of non-Euclidean geometry. It is really a pretty strong homogeneity axiom, which effectively forces the plane to look sufficiently similar around different points that it ends up (from the anachronistic view of differential geometry) having constant curvature.