Question about chart parameterization

[orion]

Suppose we have an n-dimensional manifold Mⁿ and take a coordinate neighborhood U with associated coordinate map φ: U → V where V is an open subset of ℝⁿ. So far I'm clear on this.

However, where I become confused is when some books say that φ^-1 is called a parameterization of U and basically leave it at that. I want to understand this. What exactly does a parameterization mean and how is it manifested?

The only place that I've come across parameterizations is when using parametric equations.

Could someone give a simple example showing φ and φ^-1 explicitly?

Thanks.

 

[fresh_42]

Consider $M^2 = \{ \begin{pmatrix} a && b \\ 0 && \frac{1}{a} \end{pmatrix} | \; a \neq 0 \}$.
Then all matrices of the form $\begin{pmatrix} 1+\epsilon && \delta \\ 0 && \frac{1}{1+\epsilon} \end{pmatrix}$ build an open neighborhood $U_I$ of the identity matrix.
We define $φ : U_I → V$ by $\; φ(\begin{pmatrix} 1+\epsilon && \delta \\ 0 && \frac{1}{1+\epsilon} \end{pmatrix}) = \begin{pmatrix} 1+\epsilon \\ \delta \end{pmatrix}.\; V$ here is thus an open neighborhood of $\begin{pmatrix} 1 \\ 0 \end{pmatrix}.$

Then $φ^{-1}(\begin{pmatrix} t \\ s \end{pmatrix}) = \begin{pmatrix} t && s \\ 0 && \frac{1}{t} \end{pmatrix}$.
Now given any curve in $\mathbb{R}^2$ parametrized by $s$ and $t$ gives rise to a curve in $M^2$.
You may, e.g. define a curve $γ(s,t) = \begin{pmatrix} 1+t^2 \\ 3s \end{pmatrix}$ through $\begin{pmatrix} 1\\ 0 \end{pmatrix}$ and consider its (parametrized) way in $M^2$, compute its tangent vectors and so on.

 

[orion]

Thank you for responding, fresh_42. That was instructive though I'm not sure I completely understand the example. I was thinking more in terms of the manifold being some subset of ℝⁿ rather than a space of matrices. I will have to think about that some more. It's really the only example of a manifold other than curves and surfaces that I've come even close to understanding. Thanks again.

Can anyone give an simple example of φ and φ^-1 explicitly given for a case of the manifold being a curve or a surface in ℝⁿ? I mean, really simple, even simpler than the standard circle and sphere examples in the books? Like perhaps a straight line in ℝ² or something just as simple?

 

[orion]

Actually, fresh_32, your example is sinking in and becoming very clear. I still need more examples, but I do like yours. Many thanks.

 

[fresh_42]

Well, this was an example of a very small Lie group, i.e. not so much to type in and still a Lie group which you probably will meet in more complex ways as transformation groups. If you want to calculate a vector field on a Lie group, this is the smallest non abelian example and thus least work to do but still sufficient to visualize what's going on.

Now an example of an embedded manifold is, e.g. the 2-sphere or ball in $ℝ^3.$
I'll take the sphere, for it is more obvious that you leave it as soon as you vary the parameters.

Take $S^2 = \{ (x,y,z) \in ℝ^3 \, | \; x^2+y^2+z^2 = 9\}$ and $φ: U_p → V$ where $p \in S^2 ⊆ ℝ^3$ is given by the polar coordinates $\begin{pmatrix} \frac{\pi}{4}\\ \frac{\pi}{4} \\ 3 \end{pmatrix} = \begin{pmatrix} Θ \\ Ψ \\ r \end{pmatrix}$ and $U_p$ an open neighborhood of $p$. Of course only $φ(p)$ itself is surely on the sphere with radius $r=3$, because every small variation (in $V ⊆ ℝ^3$) can leave it but still can be approximated by, e.g. the tangent space at $p$ as long as the variation is small. E.g. you could take the first few (two) terms of the Taylor series of $\cos$ and $\sin$ as an (a linear) approximation.

Here $φ^{-1} (\begin{pmatrix} s \\t \\ u \end{pmatrix}) = \begin{pmatrix} u \cdot \cos(s) \sin(t) \\ u \cdot sin(s) \sin(t) \\ u \cdot \cos(t) \end{pmatrix}$ and points on the sphere are parametrized by $s,t,u.$
In addition one can see that these parameters are only local - they repeat if you go around the sphere and thus losing their uniqueness. You might try and calculate some curves $γ(s,t,u)$ and see what happens.

Edit: IMO it might be useful to take some curves either on the sphere or in the surrounding Euclidean space (or both) and perform some calculations like differentiate them at various points or examine what is different when you take the whole ball as manifold. You could even investigate the difference between polar and cartesian coordinates and the transformation between them. All of this will probably occur in abstract treatment further ahead in your studies. (And remember that these will be local calculations in open neighborhoods.)

 

[orion]

Thank you fresh_42. That is very, very helpful. I just have a couple of questions to clear up some confusion on my part. The $\Theta$, $\Psi$, and $r$ "live" in which space? $V$? Similarly, the $s$, $t$ and $u$ also live in $V$?

 

[fresh_42]

It is all embedded in $\mathbb{R^3}$. And you can play with different kinds of manifolds here. E.g. if you fix $r=3$ you get a two-dimensional sphere $S^2$ with radius $3$ where only the angles $Θ, Ψ$ at the origin are relevant to define a point on it. I deliberately chose $3$ only for the reason that it doesn't always have to be the radius $1$ as in most examples.
However, $r ≤ 3$ would define an entire ball of radius $3$ ; $1≤r≤2$ defines a spherical shell. All of these are embedded in the three-dimensional real vector space.

Let's take the sphere, i.e. $r = u$ fixed. Then $U$ is an open set on the sphere, and $V$ an open set in $\mathbb{R^2}$ because one needs only two parameters $s,t$ to describe it. $V$ is the image of $U$ under $φ$.

But you wanted to have everything embedded so there is a third coordinate. As long as you are on the sphere, the manifold, this third coordinate is the constant radius, i.e. fixed. As soon as you investigate the sphere as part of the surrounding Euclidean space, the distance from the origin, i.e. the radius can be varied because you can leave the spehre. The parameters $s,t,u$ are nothing else than the (polar) coordinates of a point in $\mathbb{R^3}$.

$Θ, Ψ,r$ above have been taken as the polar coordinates of a special point $p$ on the sphere $S^2$.
$U$ is an open set around $p$ on this sphere.
$s,t,u$ are the parameters that describe a point in $\mathbb{R^3}$, i.e. the coordinates.
$V$ is an open set around $φ(p)$ in $\mathbb{R^3}$.

The confusion comes likely from the fact that everything "lives" in $\mathbb{R^3}$ and therefore points of the manifold and those of the surrounding space look similar. My first example with the Lie group avoided such a mess.
And this is the essential reason why we talk about manifolds instead of embedded objects: The mathematics on a manifold doesn't need an embedding!

But if you have one, you must be carefully distinguish between things going on on the manifold and those in the surrounding space. The wording parameters or parametrization is basically meant as the coordinates of points on a curve on the manifold. E.g. $γ : [0,1] → ℝ^2$ where $γ(t) = (t,1-t)$ is the parametrization (by parameter $t$) of the line segment from $(0,1)$ to $(1,0).$