Guillemin and Pollack define manifolds to be subsets of $\mathbb R^n$ locally diffeomorphic to open subsets of $\mathbb R^k$, and they define the notion of a smooth map on (not necessarily) open subset of $\mathbb R^n$ by saying that it is smooth if it locally extends to smooth maps on open subsets of $\mathbb R^n$. This definition works for manifolds (as they define it), so charts are not used as far as I can see.

In the standard definition of a manifold, the notion of a smooth map between manifolds is defined in terms of charts.

So I have two questions:

  • First, if the fact that a manifold in the sense of Guillemin-Pollack is locally diffeomorphic to $\mathbb R^k$ is not used for defining smooth maps, what for do we need it? Only to tell the dimension of a manifold? Wouldn't it be sufficient then to require that a manifold must be locally homeomorphic to $\mathbb R^k$?
  • If I define smooth maps between manifolds in the sense of Guillemin-Pollack in the usual fashion through charts, will it be equivalent definition?

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.