# Smooth Manifolds: Charts or Parametrizations?

Do I have the following correct?

If you define manifolds as subspaces of $$\mathbb{R}^N$$ as is done in Guillemin and Pollack, then you don't need charts and atlases. All you need are smooth parametrizations to describe a topological manifold as smooth. However, if you are dealing with abstract manifolds where the topological space is not a subset of $$\mathbb{R}^N$$, then you need charts with smooth "overlaps" to show a that manifold is smooth.

Here's a link that discusses why the two different characterizations of smooth are ultimately the same, Guillmin & Pollack's Definition of a Manifold. My question is more of a practical nature.

• "... then you don't need charts and atlases". I'm not sure what you could mean by this statement, because there are many applications of charts and atlases even for submanifolds of $\mathbb R^N$. – Lee Mosher Feb 1 at 17:44
• I meant "... then you don't need charts and atlases to describe a topological manifold as smooth." – TJCrow Feb 2 at 13:37

Definition: A manifold $$M$$ of dimension $$k$$ in the sense of Guillemin and Pollack is a subset $$X$$ of $$\mathbb{R}^N$$ such that for all $$x\in X$$ there is an open neighborhood $$V\subset\mathbb{R}^N$$ of $$x$$ so that $$V\cap X$$ is diffeomorphic to an open subset $$U$$ of $$\mathbb{R}^k$$. Such a diffeomorphism $$\phi:U\to V$$ is called a parametrization.
The trick here is that a diffeomorphism $$f$$ from $$X\subset\mathbb{R}^N$$ to $$Y\subset\mathbb{R}^M$$ means a smooth bijection with smooth inverse. But smooth means that, locally, $$f:X\to Y$$ always extends to a map $$F:W\to Y\subset\mathbb{R}^M$$, $$W\subset\mathbb{R}^N$$ which is smooth in the calculus 3 sense (all partials of all orders exist).
If you have a collection of parametrizations $$\phi_\lambda$$ that cover $$X$$, you have an atlas $$\lbrace \phi_\lambda^{-1}\rbrace$$. It is worth trying to show this so that you understand. The non-trivial condition that smooth maps have these local extensions is what allows us to not have to "check the overlaps". It trades one thing to check for another.