[zenorogue]

Geometry is the curvature of the fabric the space is made of.

Topology is how you sew/glue this fabric.

You can make a cylinder out of flat paper, so it is an Euclidean manifold.

You cannot make a sphere out of flat paper, so it is a non-Euclidean manifold.

Non-Euclidean geometry refers specifically to the geometry, not to the topology. It is not "anything where the distance is not the Euclidean distance" or "anything other than the Euclidean space" or "anything not related to Euclid's proof that there are infinitely many primes". Such a concept would not be useful, because it would be so broad that nothing interesting could be said about it (as you said, it would include Doom levels because of the teleports).

So a cylinder or a flat torus or a space being infinitely repeated are not non-Euclidean geometry. Monument Valley also has no relation to non-Euclidean geometry.

[nine_k]

I agree that you can glue flat paper in a way that connects disparate points without making it spherical and thus breaking the Euclidean properties: parallel lines, sum of angles in a triangle, distance, except for the teleport points. You can draw two lines parallel to a given line through a given non-teleport point: one passing through a teleport, another not passing.

Since the very notion of a line or a distance in the Mountain Valley world is ill-defined, I would not call it Euclidean either, even though each static configuration of it may be Euclidean, minus doors (which are teleports again).

[zenorogue]

Usually "non-Euclidean" is used for Riemannian manifolds, so if something is not a Riemannian manifold, it is not called Euclidean or non-Euclidean.

Just like "irrational number", which is not any number that is not rational -- we still assume that it is a real number, so i or Aleph-Null are not considered irrational.

If we include teleport points in our metric, we no longer have a manifold (if teleports are one-way, not even a metric space). Similarly for Mountain Valley, or affine/projective manifolds that can be found in some games, it is wrong to call them Euclidean or non-Euclidean.

Anyway, originally "non-Euclidean" was just the hyperbolic plane (the only geometric structure which satisfies all the Euclid's axioms except the fifth), and then (depending on whom you ask) it was extended to also mean other situations which are different but similar in nature -- so spherical geometry (which is just the opposite), Nil/Solv geometry (which also play with parallel lines), but not cylinders, affine manifolds, teleports, taxicab metric, and so on (they don't really do anything interesting to parallel lines).

[nine_k]

Thanks, that's informative!

Games on "spherical", or rather, "polar" surfaces are well-known: space flight around a gravitating body, along circular geodesics.

A shooter in a hyperbolic space, and with gravity, could be hilarious.

[zenorogue]

Unfortunately spherical geometry is often done incorrectly (in Civilization the Earth was a cylinder, in other games planets are tori).

What kind of shooter? An Asteroids-like game on hyperbolic manifolds (both 2D and 3D) can be found in HyperRogue, although there is no gravity. Gravity in hyperbolic space is a rather problematic thing, you always get some extremely weird effects, also I believe there are no stable orbits in hyperbolic spaces.