How Does a Function on Manifolds Change When Transferred from M to N?

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When transferring a real-valued function f from manifold M to manifold N, which are diffeomorphic, the function is transformed through a diffeomorphism g: M → N. This means that f can be expressed in terms of N by composing it with the inverse of g, resulting in a function that maps from N to R. The discussion highlights that diffeomorphism implies a smooth, structure-preserving map between the two manifolds, allowing for the transfer of functions. The concept of diffeomorphism is clarified, emphasizing that it does not require geometric properties on M. Ultimately, the relationship between f and N is established through the diffeomorphic mapping.
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Assume you have two manifolds M and N diffeomorphic to another. Also, there is a real-valued function f defined on M.

What happens with f when you go from M to N? How is f related to N?

thanks
 
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If you have a map from M to N does the map not also act on the function f and transform it appropriately?
 
What actually does diffeomorphic really mean?

Take a manifold M without Riemann structure defined on, i.e. without any geometric properties, so that makes it very malleable.

Is not every manifold diffeomorphic to M already M?
 
If my manifold M can be a ball, a cigar or what have you, what then is a to M diffeomorphic manifold N?
 
What happens with f when you go from M to N? How is f related to N?

Consider the diffeomorphism g:M-->N.
Then you have that the composition of g^-1 and f, maps from N to R. Is that what you are looking for?
 

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