Convert flat space function to curved space function?

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Discussion Overview

The discussion revolves around the transformation of functions defined in flat spaces to their counterparts in curved spaces. Participants explore the implications of such transformations, the concept of equivalence between functions in different geometries, and the role of metrics in these transformations.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants inquire about a general procedure for transforming functions from flat to curved spaces, questioning what constitutes an "equivalent function."
  • Others argue that points in curved spaces differ from those in flat spaces, complicating the notion of equivalence unless the curved space is homeomorphic to flat space.
  • One participant mentions the Riemann Sphere and Möbius Transformations as examples, suggesting they provide visual representations relevant to the discussion.
  • There is a question about how a point or function can discern its geometric context (flat vs. curved space) and whether a function on curved space can be expressed in terms of the curvature at each point.
  • Some participants assert that a function is defined on the set of points in a space and is unaffected by changes in the metric, while others challenge this by discussing cases where the function itself may depend on the metric, such as when considering distances between points.
  • There is a suggestion that differential changes in a function would depend on the metric, raising questions about the implications for Taylor expansions of functions in curved spaces.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between functions and the metrics of the spaces they inhabit. There is no consensus on how functions can be transformed or what constitutes equivalence between functions in flat and curved spaces.

Contextual Notes

The discussion highlights limitations in understanding how functions relate to the curvature of space and the implications of metric changes, with unresolved questions about the nature of these transformations.

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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.
 
Last edited:
friend said:
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?

How about the Riemann Sphere and Mobius Transformations...

http://www.sciencemag.org/sciext/vis2007/show/"

The movie in slide #8 of the link provides an excellent visual representation.
 
Last edited by a moderator:
slider142 said:
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.

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?
 
Your function doesn't care about the metric of the space, be it flat or curved. As I understand it, the function is defined on the set of points making up the space,and is therefore unaffected by a change of the metric.
 
Pere Callahan said:
Your function doesn't care about the metric of the space, be it flat or curved. As I understand it, the function is defined on the set of points making up the space,and is therefore unaffected by a change of the metric.

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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