philosophy : first axiom of geometry and variable curvature

Question

The very first axiom of geometry can be described as:

Two different points lay on one and only one line.

And I was wondering are there surfaces where this axiom irrecoverably fails? and I found one. (so there are infinite of them)

For example

assume the surface of a torus centered around the origin with a major radius of 3 and a minor radius of 1 symetrical in the z = 0 plane

$ x(\theta, \varphi) = (3 + 1 \cos \theta) \cos{\varphi} $

$ y(\theta, \varphi) = (3 + 1 \cos \theta) \sin{\varphi} $

$ z(\theta, \varphi) = 1 \sin \theta $

Take to points $P = (4,0,0) $ and $Q = ( 0,2,0) $ there are two lines between these two points.

(one line goes trough $ ( \sqrt{4.5} , \sqrt{4.5} , 1) $, the other line goes trough $ ( \sqrt{4.5} , \sqrt{4.5} , -1) $ )

This made me wonder:

• Does this axiom fail in all geometries with a variable curvature?

• how can you do geometry on these surfaces, without refering to some embedding structure?

• What are the philosophical consequeces of this, for example when we assume the bending of space under the influence of mass?

are there pilosophers or mathematicians who have written about this?

any info welcome

Answer 1

The most well-known case in which this axiom fails is spherical geometry where great circles play the role of "lines". Two antipodal points have an infinity of different great circles in common.

Does this axiom fail in all geometries with a variable curvature?

No. For example, cut a small disc out of the hyperbolic plane, and surround it by flat plane stretching towards infinity. That gives a kind of round-capped "pseudocone" spanning more than 360° (as seen from the flat surroundings), and any two points have exactly one geodesic between them.

In general, if the surface is homeomorphic to the usual plane and its Gaussian curvature of the surface is nonpositive everywhere, it cannot contain any nontrivial geodesic bigons.

how can you do geometry on these surfaces, without refering to some embedding structure?

Synthetic geometry a la Euclid doesn't fare well on such surfaces, but reasoning about them intrinsically is what most of differential geometry is about.

What are the philosophical consequeces of this

None that I can see.

Answer 2

There is a very closely related concept in the modern (Riemannian) geometry called NCP (No Conjugate Points); NCP in a complete simply-connected Riemannian manifold implies that any two points can be connected by a unique (minimizing) geodesic: Such geodesics can be regarded as generalizations of Euclid's notion of a line. For instance, all simply-connected complete manifolds of nonpositive curvature satisfy NCP. Few of random references: here, here, here and here. The 3rd of the links contains a proof of the following beautiful conjecture by Hopf:

Let $g$ be a Riemannian metric without conjugate points on the $n$-dimensional torus. Then $g$ is flat.

Incidentally, the following Riemannian geometry problem modeled on the Euclid's 5th postulate is open:

Suppose that $S$ is a complete simply-connected Riemannian surface of nonpositive curvature such that for each point $x\in S$ and each complete geodesic $L\subset S$ not containing $x$, there exists a unique complete geodesic $L'$ through $x$ which is disjoint from $L$. (I.e., through each point in $S$ there exists a unique geodesic parallel to $L$.) Then $S$ is isometric to the Euclidean plane.