Transforming to curved manifolds

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In summary, the conversation discusses the concept of coordinate transformations between flat and curved spaces. It is mentioned that the Riemann tensor vanishes in a flat space and no amount of transformations can go from a flat space to a curved space. The question is then posed about the possibility of a transformation from Cartesian 2D to (\theta,\phi) coordinates for a unit 2-sphere, but it is clarified that these two spaces are not topologically isomorphic. It is concluded that there is no coordinate transformation that can make one think they are on a flat plane instead of a curved sphere. However, there are transformations for the surface of a sphere that can look like R2 on a small region.
  • #1
masudr
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I know that the Riemann tensor vanishes in a flat space. And no amount of co-ordinate transformations can go from a flat space to a curved space.

Does that mean there is no transformation that will go from, say Cartesian 2D, to [itex](\theta,\phi)[/itex], the co-ordinates usually used for the unit 2-sphere? I say this since the first space is flat, and the second is curved.
 
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  • #2
Topoloically, the 2-dimensional plane is non-compact, while S^2, the 2-dimensional (surface of a) sphere is compact, so, globally, these two are not "the same", i.e., not topologically isomorphic.
 
  • #3
Thanks for your reply. I did know that, but is there no co-ordinate transformation (however exotic) that can make me think I'm on the plane instead of the sphere?
 
  • #4
No, there is no such coordinate transformation. There are, of course, transformations for the surface of a sphere that are 'approximately' flat and will look like R2 on a sufficiently small region.
 
  • #5
Many thanks, to the both of you.
 

1. What are curved manifolds?

A curved manifold is a mathematical concept that refers to a space or surface that is curved or non-linear. It can be described as a generalization of the concept of a flat surface to include surfaces that have curvature.

2. What is the purpose of transforming to curved manifolds?

The purpose of transforming to curved manifolds is to better represent the natural world, as many physical systems are inherently curved. By using curved manifolds, we can more accurately model and understand complex systems in fields such as physics, engineering, and biology.

3. How is transforming to curved manifolds useful in machine learning?

In machine learning, transforming to curved manifolds allows for more efficient and accurate processing of data. By using curved manifolds, we can reduce the dimensionality of data, making it easier to analyze and extract meaningful information from large datasets. This is particularly useful in tasks such as image and speech recognition.

4. What are some challenges in transforming to curved manifolds?

One of the main challenges in transforming to curved manifolds is the mathematical complexity involved. Curved manifolds require advanced mathematical concepts, such as differential geometry, to fully understand and work with. Additionally, transforming data to curved manifolds can be computationally expensive and time-consuming.

5. Can curved manifolds be applied to real-world problems?

Yes, curved manifolds have been successfully applied to various real-world problems, such as predicting stock market trends, analyzing climate data, and improving medical diagnoses. As our understanding of curved manifolds continues to advance, we can expect to see even more applications in various fields.

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