jokergirl

2012-Oct-16, 12:16 PM

This is a long shot but because many here are interested in the curvature of our space, I thought maybe someone could give me some good pointers. I've been doing a Wiki/WolframAlpha Walk on the subject but I can't really find the answer. Google just coughs up articles about how our (astronomical) space is curved, so either I'm not using the right search terms or it's just useless because the terms are taken up by astrophysics.

Basically, the first thing I'm looking for is if there is a standard way of describing an arbitrary curved space based on a given relationship of N points in said space*, but more importantly if there is a way of finding the most simple** curved space for a given relationship and number of points.

Right now I'm completely clueless - I'm not even sure there is a standard way of describing curvature. It's probably some set of equations, but which one? A projection of the space into flat space? A 2d curved surface can be described by translating it into 3d flat space and so on. Is that how you generally describe/visualise it? Or is there a more abstract way of describing it? (ETA: I've found Riemann tensors, not that I understand them yet.)

;)

*Yes, I am aware that basically the relationship is already a description, but the final point is actually to find the most simple space that satisfies it.

**How simple is defined is another question. As far as I can see you can either have "least amount of dimensions" or "smallest set of equations needed", or is there a way of balancing the two? That would be even cooler. But as I said, right now I'm completely clueless on that area of mathematics.

Basically, the first thing I'm looking for is if there is a standard way of describing an arbitrary curved space based on a given relationship of N points in said space*, but more importantly if there is a way of finding the most simple** curved space for a given relationship and number of points.

Right now I'm completely clueless - I'm not even sure there is a standard way of describing curvature. It's probably some set of equations, but which one? A projection of the space into flat space? A 2d curved surface can be described by translating it into 3d flat space and so on. Is that how you generally describe/visualise it? Or is there a more abstract way of describing it? (ETA: I've found Riemann tensors, not that I understand them yet.)

;)

*Yes, I am aware that basically the relationship is already a description, but the final point is actually to find the most simple space that satisfies it.

**How simple is defined is another question. As far as I can see you can either have "least amount of dimensions" or "smallest set of equations needed", or is there a way of balancing the two? That would be even cooler. But as I said, right now I'm completely clueless on that area of mathematics.