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.

 

[blue_raver22]

A parametrization is a mapping that allows us to describe a neighborhood of a point on a manifold using coordinates from a standard Euclidean space. In other words, it allows us to "parametrize" the neighborhood in terms of a set of coordinates that we are familiar with. This is useful because it allows us to use the tools and techniques from Euclidean geometry and calculus to study the manifold.

On the other hand, a coordinate system is the inverse mapping of a parametrization. It takes a point on the manifold and maps it to a point in the standard Euclidean space. This allows us to express any point on the manifold in terms of coordinates that we are familiar with.

In the big picture of manifolds, parametrizations and coordinate systems play a crucial role. They allow us to define and study manifolds using tools and techniques from Euclidean geometry and calculus. This makes the study of manifolds more accessible and allows us to make connections between different manifolds by using the same coordinate systems. Additionally, parametrizations and coordinate systems are essential for defining and understanding concepts such as tangent spaces, tangent bundles, and differential forms on manifolds.