Do diffeomorphisms have to be one-to-one functions?

Therefore, for a diffeomorphism, the original function must be one-to-one to ensure its inverse is a function. However, the inverse can also be a relation, but it may not have all the properties of a function. In summary, for a diffeomorphism, the original function must be one-to-one to guarantee its inverse is a function, but it can also be a relation.
  • #1
Antineutrino
6
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The definition of a diffeomorphism involves the differentiable inverse of a function, so must the original function be one-to-one to make its inverse a function, or can the inverse be a relation and not a function?

Sorry if it's a silly question, I am just a second semester calc student who looked at this for fun.
 
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  • #2
Antineutrino said:
The definition of a diffeomorphism involves the differentiable inverse of a function, so must the original function be one-to-one to make its inverse a function, or can the inverse be a relation and not a function?

Sorry if it's a silly question, I am just a second semester calc student who looked at this for fun.
If a function is not one-to-one, then it has no inverse.
 

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