1. ## Question about local parameterization of a manifold

Just to lay the backdrop, suppose that I have an n-dimensional manifold $M^n$ and choose a particular coordinate chart on the manifold $(U,\phi)$, $\phi: U \rightarrow \mathbb{R^n}$.

I understand that $\phi$ is the local system of coordinates. My question concerns $\phi^{-1}$. In many books or lecture notes, $\phi^{-1}$ is called a local parameterization of $M$, and it is left at that. I am having trouble envisioning how the inverse map of a coordinate map is a parameterization.

Would someone please walk me through a simple example to clear up my confusion as to $\phi$ and $\phi^{-1}$? The thing that makes the confusion worse is that I somehow think that it should be obvious and that I already know this from coordinate transformations I have done in calculus and in physics.

Thanks.

2. ## Re: Question about local parameterization of a manifold

Hey weirdanalysis.

Are you familiar with an inverse operator in linear algebra? If you are then it will make a lot more sense of what this inverse mapping will mean when extended to multiple dimensions.

3. ## Re: Question about local parameterization of a manifold

Originally Posted by chiro
Hey weirdanalysis.

Are you familiar with an inverse operator in linear algebra? If you are then it will make a lot more sense of what this inverse mapping will mean when extended to multiple dimensions.
Until just this moment I thought I was but now I'm not so sure because I can't see what you're driving at. I mean, I know what an inverse map is and I'm comfortable even with it being multidimensional (I think). But here again, I feel like I should know this ... like I'm at the threshold of understanding but I just can't quite get there.

4. ## Re: Question about local parameterization of a manifold

What it means for a linear operator is that you have a matrix (call it A) and if you are mapping a vector X to B then the inverse operator will be A_inverse.

If it is an n-dimensional system you need an n-dimensional operator and the operator need not be linear (i.e. not be represented solely as a matrix - but often a function of matrices).

That's the basic idea.

5. ## Re: Question about local parameterization of a manifold

I am having trouble envisioning how the inverse map of a coordinate map is a parameterization.

Practice on the unit circle $\mathbb{S}^1\subseteq \mathbb{R}^2$.
Construct a local parametrization (say $x=cosu,y=sinu, 0\leq u\leq \pi/2$) and inverse it, to obtain $(x/\sqrt(x^2+y^2),y/\sqrt(x^2+y^2))$.

Do note that many writers choose to define local charts using $\phi^{-1}$ instead of $\phi$.