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## Main Question or Discussion Point

Is there a general procedure to convert or transform a function that is defined on a flat space into an equivalent function in curves spaces?

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Is there a general procedure to convert or transform a function that is defined on a flat space into an equivalent function in curves spaces?

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What do you mean by "equivalent function" ? Since the points on the curved space will be different from the points in the flat space, unless the curved space is just extrinsically curved or is just a portion of a curved space homeomorphic to the flat space.

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How about the Riemann Sphere and Mobius Transformations...Is there a general procedure to convert or transform a function that is defined on a flat space into an equivalent function in curves spaces?

http://www.sciencemag.org/sciext/vis2007/show/" [Broken]

The movie in slide #8 of the link provides an excellent visual representation.

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I guess that's my question. How does a point know if it is in flat space or curved space? And therefore, how does a function know if it is in flat or curved space? Can a function on curved space be expressed in terms of the curvature of that space at each point? And can this be converted to some function on flat space coordinates?What do you mean by "equivalent function" ? Since the points on the curved space will be different from the points in the flat space, unless the curved space is just extrinsically curved or is just a portion of a curved space homeomorphic to the flat space.

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What if the function is the metric? I suppose if the function depends on two points, or is it the distance between two points, then it depends on the curvature of the space. Is this right? What about differential changes in the function? That differential depends on the metric, right? So a Taylor expansion used to represent the function WOULD depend on the metric. I guess the question still remains for me. Thanks.

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