Why don't we have many non euclidean geometries out there?

Question

In the current semester I've taken a course about Non-Euclidean Geometry. During the course, we presented two types of non-euclidean geometries: the spherical geometry and the hyperbolic geometry. So that makes three distinct geometries, along with the familiar euclidean one.

However, it seems we weren't really taught what exactly a geometry is - what are our expectations from something we'd like to call "a geometry"?

If I were asked to answer this question, I would say that a geometry must have notions of point, line, angle, distance, length and area.
But in another course- Differential Geometry - we've learned about riemannian metrics on open subsets of the plane, which endow the set with all the mentioned notions (distance being the infimum of lengths of curves connecting two points, lines being geodesics...).
So, if I take an arbitrary riemannian metric, and look at the geometric notions it induces, what deprives this structure from being called a geometry? What is so special about the three that we've been introduced?

I'd be happy if you try to include intuition and motivation as a part of your answers. Thanks!

Answer 1

I think the key notion here is curvature : https://en.wikipedia.org/wiki/Gaussian_curvature

Euclidean geometry is the geometry with zero curvature, hyperbolic geometry is the geometry with negative curvature, and elliptic or spherical geometry is the geometry with positive curvature.

Of course any Riemannian metric gives rise to geometric questions (behaviour of the geodesics, etc) but in practice it is hard to get general results in the situation where the curvature has non-constant sign (and of course a natural approach is to try and decompose such a Riemannian manifold into hyperbolic, euclidean and spherical parts, which also explains why those three are so important).

Edit (too long for a comment)

Warning: I am not an expert in these questions.

I think a bit of caution is needed regarding the interplay of Euclid's axioms and modern geometry. I looked up Euclid's axioms on wikipedia, and here are the first two:

[It is possible]

1. "To draw a straight line from any point to any point."

2. "To produce [extend] a finite straight line continuously in a straight line."

The modern definition of straight line is geodesic. Those two properties are satisfied if and only if the manifold is geodesically complete https://en.wikipedia.org/wiki/Geodesic_manifold So when translating Euclid's axioms, I guess we mean them to not apply to any manifold with constant curvature but at least only to the complete ones. Regarding your question in the comments, I would suggest that the Killing-Hopf theorem (https://en.wikipedia.org/wiki/Killing%E2%80%93Hopf_theorem) might be an answer: there are only 3 complete simply connected Riemannian manifolds with constant curvature.

Answer 2

I think most people would say that a Riemannian manifold is a "geometry" in some sufficiently broad sense. What it is not, is a geometry that can be easily reasoned about axiomatically: you generally have to actually get your hands dirty with coordinates.

On the other hand, spherical and hyperbolic geometries are "non-Euclidean" in the sense that they don't obey Euclid's axioms, but they still are characterized by some set of axioms, which doesn't look too different from a set of axioms characterizing Euclidean geometry. And frequently (as in the title of your class, it looks like), "non-Euclidean geometry" is used to refer to the study of such things. That is, it would be more accurate to call it "non-Euclidean axiomatic geometry" or some such.

Answer 3

Eventually I found a satisfying answer (in my opinion) in the book "Visual Complex Analysis", so I decided to post it here in my own words as an answer for those of you who are interested.

The answer extends the previous answer given here, that mentions the notion of curvature. The three geometries (euclidean, shperical, hyperbolic) are unique, and differ from an arbitrary "riemannian geometry", by having the same gaussian curvature at any point (euclidean: $K=0$, shperical: $K>0$, hyperbolic: $K<0$). These are "the geometries of contant curvature".
Note that gaussian curvature is indeed a property of the riemannian metric and the intrinsic geometry alone, due to Theorema Egregium.

The natural question that arises is, why would a contant $K$ be crucial for having a nicely structured geometry. The answer lies in another answer to the question of "what is geometry?".
My initial answer, as you can read, was "a structure that deals with notions of point, line, angle, distance, length and area." But Felix Klein had a more precise characterization: geometry is "the study of entities that are invariants of isometries." [this is not a quote.]
Note that all the notions I mentioned are inculded in his definiton.

Now, here's the main point: by Theorema Egregium, the curvature at a point can be induced by knowing only the geometric structure near that point. Therefore if an isometry maps one point to another, they must have the same $K$.
So in an arbitrary riemannian geometry, the group of isometries would look quite confined, incomplete, unnatural - while the isometries groups of our three special geometrie have a much nicer and more natural structure, yielding a nice and natural geometric sturcture.

So, summarizing the last two paragraphs in one sentence, we can say that the non-euclidean geomtries are those who "look the same from any point of view" - this is the key property for their natural structure.