Parametrization vs. coordinate system

[lmedin02]

I am reading Differential Topology by Guillemin and Pollack.

Definition: X in R^N is a k-dimensional manifold if it is locally diffeomorphic to R^k.

Suppose U is an open subset of R^k and V is a neighborhood of a point x in X.
A diffeomorphism f:U->V is called a parametrization of the neighborhood V.

The inverse mapping f^-1:V->U is called a coordinate system.

Why is f a parametrization and its inverse a coordinate system? How do these terms fit in the big picture of manifolds?

I understand that we are rewriting V in X into a new coordinate system in R^k, that is easier to work with as oppose to some abstract space X. I not to sure of how to interpret the parametrization f.

 

[torquil]

Think of it as follows:

By coordinate map one means a mapping that specifies coordinate values for each point in your manifold M. So it would go in the direction M -> R^m

Think of a path in M parametrized by a real number. It is represented as a mapping in the direction R -> M. Similarly, a parametrization of a surface in M could be a mapping R^2 -> M. A parametrization of a whole m-dimensional set in M would be a mapping of the type R^m -> M, exactly opposite to the coordinate mapping.

I didn't bother to specify subsets etc. above, just wanted to illustrate why a parametrization is a mapping from some R^m into M. But maybe you already understood this, and wondered about something else? Sorry about not using your terminology, I just wrote like I'm used to.

Torquil

 

[lmedin02]

In my reading I got confuse with this definition:

Definition: X in R^N is a k-dimensional manifold if it is locally diffeomorphic to R^k; meaning that each point x\in X possesses a neighborhood V in X which is diffeomorphic to an open subset U of R^k. Is it necessary for U to be open?

If U was not open, we can use a local extension to define a smooth map?

 

[zhentil]

Neighborhoods are by definition open. It couldn't be homeomorphic to anything but an open subset of R^k.