Convert flat space function to curved space function?

[friend]

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?

 

[slider142]

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.

 

[Ynaught?]

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.

 

[friend]

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?

 

[Pere Callahan]

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.

 

[friend]

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.