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

[Icosahedron]

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

 

[Eidos]

If you have a map from M to N does the map not also act on the function f and transform it appropriately?

 

[Icosahedron]

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?

 

[Icosahedron]

If my manifold M can be a ball, a cigar or what have you, what then is a to M diffeomorphic manifold N?

 

[Cincinnatus]

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?