Embedding manifolds that are not very flat.

In summary, it has been stated on these forums that a small piece of relatively flat 4D spacetime from General Relativity can be embedded in ten dimensions. However, if the small piece of spacetime is not very flat, it may require a lower number of embedding dimensions, possibly six, to properly map the curved 3D piece. Additionally, for GR, there may be constraints that lower the Nash lower bound for compact manifolds.
  • #1
Spinnor
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I think I read on these forums that a small piece of relatively flat 4D spacetime of General Relativity can be embedded in ten dimensions. What happens if the small piece of spacetime is not very flat, does this change the number of embedding dimensions required?

Thanks for any help!
 
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  • #2
Spinnor said:
I think I read on these forums that a small piece of relatively flat 4D spacetime of General Relativity can be embedded in ten dimensions. What happens if the small piece of spacetime is not very flat, does this change the number of embedding dimensions required?

Thanks for any help!

I am not sure what you mean by embed since space time has a metric of siganture -1. Can you elaborate?
 
  • #3
wofsy said:
I am not sure what you mean by embed since space time has a metric of siganture -1. Can you elaborate?

Thanks, I'll try. Leave time out. We can embed a 2 dimensional sphere in 3 dimensions. We can embed a 3 dimensional sphere in 4 dimensions. If we have some small piece of some arbitrary 3 dimensional curved manifold my guess is we need a flat space of at most 6 dimensions to properly map the small curved 3 dimensional piece, is that close?

Thanks for any help!
 
  • #4
Spinnor said:
Thanks, I'll try. Leave time out. We can embed a 2 dimensional sphere in 3 dimensions. We can embed a 3 dimensional sphere in 4 dimensions. If we have some small piece of some arbitrary 3 dimensional curved manifold my guess is we need a flat space of at most 6 dimensions to properly map the small curved 3 dimensional piece, is that close?

Thanks for any help!

Space like sections of Minkowski space are flat. They are isometric to flat Euclidean three space. If curvature is added then there is a general isometric embedding theorem of Nash for compact manifolds. For GR there may be constraints that lower the Nash lower bound but I have no idea. My naive sense is that space is only slightly not flat and then only in the near vicinity of large masses.
 
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Related to Embedding manifolds that are not very flat.

1. What are embedding manifolds?

Embedding manifolds are mathematical objects that are used to represent curved surfaces or spaces in a flat, Euclidean space. They are commonly used in fields such as differential geometry, topology, and computer graphics.

2. What does it mean for a manifold to be "not very flat"?

A manifold is considered "not very flat" if it has significant curvature, meaning that the distance between two points on the surface is not equal to the straight line distance between those two points in a flat, Euclidean space.

3. Why is it important to study embedding manifolds that are not very flat?

Studying embedding manifolds that are not very flat allows us to better understand the structure and properties of curved spaces, which have many real-world applications. For example, they are used to model the shape of the Earth's surface, understand the behavior of light in the universe, and design computer graphics and animations.

4. How are embedding manifolds that are not very flat represented mathematically?

Embedding manifolds are represented using mathematical equations and formulas, such as the Gaussian curvature equation, which describes the curvature at a particular point on a surface. These equations can be used to calculate properties of the manifold, such as its curvature, geodesic paths, and surface area.

5. What are some challenges in embedding manifolds that are not very flat?

Embedding manifolds that are not very flat can be challenging due to the intricate and complex nature of curved spaces. It requires advanced mathematical techniques and computational methods to accurately represent and analyze these manifolds. Additionally, there may not be a single, unique way to embed a non-flat manifold, leading to different representations and interpretations of the same space.

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