Increasing the dimensions of a manifold

In summary, the conversation discusses the possibility of eliminating curvature in a Riemannian manifold with a metric by adding an additional dimension and extracting all the curvature information into it. This would result in a flat R^3 chart, but would also change the topological property of the manifold. The feasibility of this approach is uncertain.
  • #1
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Suppose I have a R^3 manifold that goes into R^3 charts, if that is possible. The manifold has curvature and is Riemannian and has a metric. I want to eliminate all curvature in R^3 charts, so I want to add another dimension to the manifold, I would extract all the curvature information from the manifold and deposit it into this new 4th dimension. I would probably also have to add some kind of topology or map to the new dimension describing the curvature. But I would have flat R^3 chart. Is this possible at all?
 
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  • #2
I'm not sure what your question is exactly, but the number of dimensions is a fundamental topological property of a manifold, so if we change the number of dimensions we have a different manifold.
 

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