Let M , N be smooth manifolds, and let F : M → N be a smooth immersion. I know that F is a local embedding, but is it also an embedding almost everywhere?In other words, does there exists a set of measure zero X ⊂ M such that F restricted to M ∖ X is a smooth embedding?

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Let   M  ,  N be smooth manifolds, and let   F  :  M  →  N be a smooth immersion. I know that   F is a local embedding, but is it also an embedding almost everywhere?In other words, does there exists a set of measure zero   X  ⊂  M such that   F restricted to   M  ∖  X is a smooth embedding?

Odin Jacobson · Accepted Answer

No: a covering map of smooth manifolds is an immersion but (I&#039;m nearly sure) never has this property unless it is a diffeomorphism.For example, map       S    1    =  {  z  ∈      C    ∣      |    z      |    =  1  } to itself by   z  ↦      z    2  . This is an immersion but a measurable subset   T of the domain on which the map is injective has measure at most       1    2   the measure of       S    1  , because if   A  :      S    1    →      S    1   is the antipode (  A  (  z  )  =  −  z) then   T  ∩  A  (  T  )  =  ∅, so that   λ  (  T  )  +  λ  (  A  (  T  )  )  ≤  λ  (      S    1    ), where   λ is the length measure on       S    1  ; but   λ  (  T  )  =  λ  (  A  (  T  )  ), since   A is an isometry.

Let M , N be smooth manifolds, and let F : M →<!--

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