3

It is known that there can be different smooth structures on the real line, but they are all diffeomorphic to each other. So, when a theorem says that the smooth structure of a certain manifold is uniquely determined, does it mean merely up to diffeomorphism or that the smooth structure is really unique?

For example, it is known that the smooth structure of an embedded submanifold is unique. Is it only unique up to diffeomorphism? (Page 114 of Lee's Smooth Manifolds)

Chung Ng
  • 41

1 Answers1

4

Unique up to diffeomorphism. To see this, note that every non-empty positive-dimensional topological manifold which admits a smooth structure admits uncountably many distinct smooth structures; see problem 1-6 of Lee's Introduction to Smooth Manifolds (second edition).

Michael Albanese
  • 99,526