# What is the geometry of light cones if space is curved/non-Euclidean?

In light cone diagrams, the plane corresponding to the present is always the Euclidean one, but what if space is curved? Now, I've also seen diagrams where spacetime is supposed to be regarded as curved and there are many standard Minkowskian light cones all pointed in different directions that are all supposed to exist on the tangent bundle of spacetime or whatever, I've also seen ones where the future of the light cone is bent towards a blackhole, etc. None of that is what I'm looking for. First of all I want to reduce the scope of all Riemannian geometries to that of the classical geometries (Euclidean, elliptical, hyperbolic) since, as far as we know, one of these describes the large-scale geometry of the universe.

What I want to know is 1) how do we rigorously describe the hypersurfaces of constant quadrance (referring to whatever the generalization of the Minkowski quadratic form may be)? What I mean here is that I'm not just curious about double cones, but hyperboloids as well. 2) How can we visualize these hypersurfaces in relation to an observer, i.e. create hyperbolic/elliptical versions of the classical light cone diagram. It's really the first question I'm more interested in since I'm sure I can contrive a way to answer the second should I receive more information about the first.

Here's what I've thought about thus far:

In the elliptical version I think of spacetime as a 4-dimensional cylinder, specifically $S^3 \times \mathbb{R}$. This means that to allow myself to visualize it I will instead think about a 2D cylinder. The light cone then corresponds to a pair of lines going around the cylinder that make $\frac{\pi}{4}$ angles with the cylinders axis of symmetry. The "volume" encompassed by a cross section of the cone will increase proportionally to the amount of time that has passed to the power of the space dimension. The lines wrapping around the cylinder corresponds to the fact that, if you shoot a laser in a direction in a spherical universe, it will eventually come back and hit you. I feel as though this case is rather easy to grasp.

Now the hyperbolic case is quite a bit more fascinating, how are should we replace the plane that represents the present in the normal diagram? With a disk? With half a plane? With a hyperboloid? It seems like it would be very difficult to a light cone in the scene for all of these options. We know that the volume encompassed by the cone increases exponentially as time passes, but that doesn't tell us very much in the way of visualizing it. Think about a 2+1 dimensional space time in which a radial pulse of light, $A$, is broadcast at the same time as another pulse, $B$. Assuming there is a very large number of photons, there should be at exactly one photon from $B$ whose trajectory is parallel to some unique photon in $A$ (corresponding to the fact that a double cone is the revolution of a line). In the hyperbolic case, would there perhaps be multiple photons from $B$ that are parallel to the one in $A$? This speculation is due to the replacement of the parallel postulate in hyperbolic geometry. It would it would necessarily mean that each parallel photon in $B$ would have multiple parallel photons in $A$, which I think would imply that there are photons in $A$ that are parallel to each other, which doesn't make much sense since they all came from the same source. I'm not really sure how to approach this.