Is there a surface in Euclidean space that admits elliptic geometry?

Question

As I understand, on a pseudosphere, a surface of constant negative curvature, we can realize a part of the hyperbolic plane (but not the entire plane due to Hilbert's 1901 theorem) and use this for example to prove that hyperbolic geometry is just as consistent as Euclidean geometry. But how does this compare to elliptic two-dimensional geometry? Naturally, we can consider a sphere, which is a surface of constant positive curvature, but a sphere only admits spherical geometry and not elliptic geometry if we consider Euclid's postulates (for example there is not always a unique shortest line between two points). How can we then use the sphere to show that elliptic geometry is as consistent as Euclidean geometry? Does there exist some other surface which we can use to do so? Can someone clarify this for me?

Answer 1

Here is an extended version of my comments.

1. Note that Euclid did not have a complete set of axioms in the modern sense; Hilbert was the first to give such a list. If one takes Hilbert's axioms and substitutes the existence of a parallel line axiom with the nonexistence of a parallel line axiom ("For every line $L$ and a point $p\notin L$ there is no line through $p$ which is disjoint from $L$") then one can show that the resulting geometry is isomorphic to the one of the real projective plane equipped with homogeneous Riemannian metric (you have the freedom to choose the constant curvature, but that's all). Then you realize that real projective plane does not embed in $R^3$ for topological reasons (but admits an isometric $C^1$-smooth immersion by Nash-Kuiper's theorem).

2. As for Beltrami's work: Consistency of a geometry from (post) Hilbert viewpoint has nothing to do with existence of an (isometric) embedding in a particular space. For instance, for a Riemannian manifold to exist, it suffices to define it via an atlas of charts together with a Riemannian metric tensor. What Beltrami did was to embed isometrically a proper open subset of the hyperbolic plane in $R^3$. Since a proper open subset of $H^2$ violates axioms of hyperbolic geometry (it is not homogeneous!), existence of such an embedding does not prove consistency of hyperbolic geometry. (Note that Hilbert proved that $H^2$ does not admit a $C^2$-smooth isometric embedding in $R^3$.) What Beltrami (or Poincare, depending on what you read) realized is that you can simply write down the familiar expression for the hyperbolic metric tensor on the upper half-plane (or the unit disk) in order to establish existence of hyperbolic geometry. Lastly, if you really want to embed isometrically all these geometries (including the elliptic one) in some Euclidean space, just use Nash isometric embedding theorem. You can embed all 2-dimensional geometries in $R^{10}$ via $C^\infty$-smooth isometric embeddings as a corollary of Gunther's improvement on Nash's theorem (I am sure that 10 is still not optimal, I think, it can be reduced to 6): M. Günther, "Zum Einbettungssatz von J. Nash", Mathematische Nachrichten 144 (1989) p. 165-187.

Answer 2

One common way to realize elliptic geometry is by taking the sphere but identifying antipodal points. So you say a point and the point directly opposite the first one are in fact one and the same in terms of your model. You use two spherical points to model a single elliptic one.

for example there is not always a unique shortest line between two points

The case where this breaks is if the two points you try to connect are in fact antopodal. In the above model, these would be considered the same point, and the axioms agree that there is no unique line connecting a point to itself. So at least this example works out just fine, and other problems will get resolved in a similar manner.

Most people read Euclid's axioms in such a way that elliptic geometry violates not only the fifth, but also the one about how you can infinitely extend a line. So in that sense, hyperbolic geometry is consistent with the first four axioms, while elliptic geometry is not. But this depends in part on how exactly you formalize Euclid's axioms.

If you follow the axiomatization used by Hilbert in his Grundlagen der Geometrie (Foundations of Geometry), then the part about extending a line segment, which would be axiom IV.1 in Hilbert's work, seems valid enough. But some of the axioms of order, axiom II.3 in particular, are violated. Between-ness on the sphere simply isn't the same as in the Euclidean or hyperbolic plane. So even in Hilbert's axiomatization, elliptic geometry violates more than just the axiom of parallels.