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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.

ToSeek
2012-Oct-16, 02:41 PM
Borderline call, but moved from OTB to Q&A as a place more likely to get appropriate responses.

caveman1917
2012-Oct-16, 03:28 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.

Are you looking to describe general curved surfaces as they are used in physics or as a more general mathematical concept? The difference is that surfaces used in physics usually take more structure on the surface so you'd have to look into Riemannian geometry. If you're looking for something more general than take a look at differential geometry or even differential topology.


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.

The answer depends on how those relationships will be formed. For example if a relationship between two points says something about their distance you'll need a metric on your surface, if it says something about an angle you'll need an inner product. If it says something like that one is contained in an open subset around the other than you need some topological considerations. There are standard ways for doing all of that.

jokergirl
2012-Oct-16, 06:56 PM
More general, then. Basically I want to find the simplest (whatever that means) space in which N points can be mutually equidistant for a given N.
In flat space it's simple, N=dimensions+1. If you curve space, you should be able to get away with less dimensions, but how to describe the curvature you need for a given N or dimension?

And no, don't ask me why. I don't even think it's useful. Pure idle interest.

;)

jfribrg
2012-Oct-16, 07:43 PM
If you are looking for distances, then a good search term would be "metric space", which should give you plenty of resources involving topology. As you alluded to with your "whatever that means" comment, there are different ways to measure distance. The most familiar one is the Euclidian distance, but there are plenty of others that you might want to look at. Depending on how you define the distance, the associated space will have different properties. In general a distance function is any function that takes two points in space as an input and outputs a non-negative real number that tells you how far apart the two numbers are. This function must never return a negative number, is zero only when the two input points are the same, and satisfy the triangle inequality (which in the Euclidian distance case states that the hypotnuse of a triangle can not be longer than the total length of the other two sides). Once you decide on your distance metric, many of the properties of the space kind of fall in place, but it's possible that you don't need these properties to find your equidistant space. Maybe you can take your N points and use these to define a Bezier curve or something similar and output a custom distance function in which the points are guaranteed to be exactly equidistant.

Ivan Viehoff
2012-Oct-17, 09:07 AM
If you are looking for distances, then a good search term would be "metric space", which should give you plenty of resources involving topology. As you alluded to with your "whatever that means" comment, there are different ways to measure distance. The most familiar one is the Euclidian distance, but there are plenty of others that you might want to look at. Depending on how you define the distance, the associated space will have different properties. In general a distance function is any function that takes two points in space as an input and outputs a non-negative real number that tells you how far apart the two numbers are. This function must never return a negative number, is zero only when the two input points are the same, and satisfy the triangle inequality (which in the Euclidian distance case states that the hypotnuse of a triangle can not be longer than the total length of the other two sides). Once you decide on your distance metric, many of the properties of the space kind of fall in place, but it's possible that you don't need these properties to find your equidistant space. Maybe you can take your N points and use these to define a Bezier curve or something similar and output a custom distance function in which the points are guaranteed to be exactly equidistant.
Whilst it is possible to consider the curvature of a metric spaces, eg, http://math.berkeley.edu/~lott/zlott.pdf a metric space alone doesn't really give enough machinery to get very far. Which is why that lecture presupposes knowledge of differential geometry. Metric spaces are normally discussed in relation to their topological properties. For example, the Wikipedia article on metric spaces mentions only topological properties.

There are two approaches to curvature. One is to consider a space embedded into a higher dimensional Euclidean space. For example, a we can consider a line in 2-d space, like a circle. It has a certain curvature. Or a line that must be embedded into 3-d spaces, like a spring. It has a more complex curvature than a plane curve. There are lines that sit in 4 but not 3 dimensions with still more complex curvature, etc. In the same way, we can take a 3-dimensional subset of Euclidian n-space (n>3) and consider its curvature. Or if we wish to consider the mutual curvature of space and time, a 4-d subsent of Euclidean m-space (m>4). These are rather more difficult to get your head around.

The more general approach is not to embed into another space, but rather consider the inherent curvature of the space itself without an embedding. This is necessary because there are curvatures that will not embed into Euclidean n-space. Now we need to consider more general manifolds. A manifold is a space which is locally similar to Euclidean n-space. The study of manifolds lies within the discipline of differential geometry. You will encounter Riemannian metrics and Riemannian manifolds, and discover that a Riemannian manifold is indeed a metric space. You will probably stumble over a lot of tensors.

Unfortunately this is a rather tricky area of maths and there is no getting away from it.

jokergirl
2012-Oct-18, 07:53 AM
Thank you for those hints! I'll get a little bit further with them, I hope :)