Local parameterizations and coordinate charts

["Don't panic!"]

I have recently had a lengthy discussion on this forum about coordinate charts which has started to clear up some issues in my understanding of manifolds. I have since been reading a few sets of notes (in particular referring to John Lee's "Introduction to Smooth Manifolds") and several of them have mentioned about the inverse map of a coordinate map as giving a local parametrization to a point in a given patch on a manifold. By this is it meant that, given an n-dimensional manifold M and a homeomorphism \phi :U\subset M\rightarrow V\subset\mathbb{R}^{n} from a patch on the manifold U\subset M, then we can parametrize a point p\in U via the inverse map \phi^{-1}:V\subset\mathbb{R}^{n}\rightarrow U\subset M. More explicitly, if \phi (p)=(x^{1},\ldots ,x^{n}) are the coordinates of p in \mathbb{R}^{n}, then p=(\phi^{-1}\circ\phi )(p)=\phi^{-1}(\phi(p))=\phi^{-1}(x^{1},\ldots ,x^{n})=(u^{1},\ldots ,u^{n}) where (u^{1},\ldots ,u^{n}) is the local parametrization of p on M, with u^{i}=u^{i}(x^{1},\ldots ,x^{n}) are functions whose domain is \mathbb{R}^{n}.
If so, what really is the difference between parametrizations of points and their corresponding coordinates?

If I have understand this notion of parametrization correctly, then is the following discussion correct? If we take the example of a 2-sphere S^{2}\subset\mathbb{R}^{3} (considering it as a subset of \mathbb{R}^{3}, i.e. essentially embedded in \mathbb{R}^{3}), then is the 2-tuple (\theta , \phi)\in\mathbb{R}^{2} the coordinates of a point on the manifold (with the mapping defined by p\mapsto (\theta , \phi)) and its corresponding local parametrization on the manifold, (\sin (\theta)\cos (\phi), \sin (\theta)\sin (\phi), \cos (\theta))\in S^{2}\subset\mathbb{R}^{3} (with the inverse mapping defined by (\theta , \phi)\mapsto (\sin (\theta)\cos (\phi), \sin (\theta)\sin (\phi), \cos (\theta))) ?

From reading John Lee's books on smooth manifolds and Riemannian geometry (and from a previous discussion on here), I think it is correct to say that (when a metric is defined on the manifold) one can only use Cartesian coordinates to label points in a patch on a manifold if the curvature of the manifold is zero (i.e. it is "locally flat") as then there will exist a local isometry between the manifold between the manifold and flat Euclidean space. Mathematically, if (M,g) is locally flat (i.e. has vanishing local curvature) then there will be an isometry \psi to an open set in (\mathbb{R}^{n},\bar{g}) (where g is the metric defined on the n-dimensional manifold M, and \bar{g} is the Euclidean metric defined on \mathbb{R}^{n}).

 

[WWGD]

I am not sure I understand your question, but for one, your manifold can be treated, locally, as a copy of $\mathbb R^n$, up to homeomorphism, but not up to isometry. All local topological properties of $\mathbb R^n$ are satisfied in any given patch. In some cases, but not always local properties (like existence of a Riemannian metric) can be patched together.

 

["Don't panic!"]

Yes sorry, it is a rather long-winded question. More explicitly,

1. What is the difference between parametrization of points in a patch on the manifold and coordinates in $\mathbb{R}^{n}$?

2. Is the existence of a homeomorphism, $\phi : U\subset M\rightarrow V\subset\mathbb{R}^{n}$ from a patch on the manifold to a subset of $\mathbb{R}^{n}$ literally a statement that we can use coordinates in $\mathbb{R}^{n}$ locally to parametrize points in a patch on the manifold. In other words, in practice do we use $\phi$ to assert that a particular local coordinate system exists and then use its inverse map $\phi^{-1}: V\subset\mathbb{R}^{n}\rightarrow U\subset M$ to label points on the manifold (as in the 2-sphere example that I gave in my first post)?

3. I have read in John Lee's book on Riemannian geometry that a manifold is locally flat (has zero curvature locally) if there exists a local isometry between $(M,g)$ and $(\mathbb{R}^{n},\bar{g})$ (where $g$ is the metric on $M$ and $\bar{g}$ is the standard Euclidean metric. Does this imply that one can only use Cartesian coordinates (e.g. $(x,y,z)\mapsto (x,y,z)$, where $(x,y,z)\;\in\mathbb{R}^{3}$) when the local curvature is zero (i.e. where Euclidean geometry holds, e.g. the parallel postulate)?

 

[WWGD]

3) No, I don't think so, a cylinder is flat (tear it open and spread it over a plane) , but it is not necessary ( nor even possible) to describe it using global Cartesian coordinates.

 

["Don't panic!"]

3) No, I don't think so, a cylinder is flat (tear it open and spread it over a plane) , but it is not necessary ( nor even possible) to describe it using global Cartesian coordinates.

But it is possible to describe by local Cartesian coordinates (or "local Euclidean coordinates", as technically, according to John Lee, a Cartesian coordinate system consists of maps $x^{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ such that if $(a^{1},\ldots ,a^{n})\in\mathbb{R}^{n}$ then $(a^{1},\ldots ,a^{n})\mapsto x^{i}(a^{1},\ldots ,a^{n})=a^{i}$), right? (Isn't the Cartesian coordinate system simply an identity map from real $\mathbb{R}^{n}$ into $\mathbb{R}^{n}$? If this is the case, then it makes sense to me that one can only use Cartesian coordinates (or Euclidean flat space coordinates when the space you are considering is $\mathbb{R}^{n}$, or at least isometric to it?!)

Are you able to shed any light on the first two questions at all?

 

[HallsofIvy]

Yes sorry, it is a rather long-winded question. More explicitly,

1. What is the difference between parametrization of points in a patch on the manifold and coordinates in $\mathbb{R}^{n}$?

2. Is the existence of a homeomorphism, $\phi : U\subset M\rightarrow V\subset\mathbb{R}^{n}$ from a patch on the manifold to a subset of $\mathbb{R}^{n}$ literally a statement that we can use coordinates in $\mathbb{R}^{n}$ locally to parametrize points in a patch on the manifold. In other words, in practice do we use $\phi$ to assert that a particular local coordinate system exists and then use its inverse map $\phi^{-1}: V\subset\mathbb{R}^{n}\rightarrow U\subset M$ to label points on the manifold (as in the 2-sphere example that I gave in my first post)?

These first two questions are essentially the same. The homeomorphism, \phi assigns to any point in U a point in Rⁿ so assigns to any point in U n numbers, the "parameterization" or "coordinates" for that point.

3. I have read in John Lee's book on Riemannian geometry that a manifold is locally flat (has zero curvature locally) if there exists a local isometry between $(M,g)$ and $(\mathbb{R}^{n},\bar{g})$ (where $g$ is the metric on $M$ and $\bar{g}$ is the standard Euclidean metric. Does this imply that one can only use Cartesian coordinates (e.g. $(x,y,z)\mapsto (x,y,z)$, where $(x,y,z)\;\in\mathbb{R}^{3}$) when the local curvature is zero (i.e. where Euclidean geometry holds, e.g. the parallel postulate)?

It implies that one can use Cartesian coordinates, but not that one can only use Cartesian coordinates. An obvious example is R³ itself where you can use Cartesian coordinates, spherical coordinates, cylindrical coordinates, etc.

 

["Don't panic!"]

These first two questions are essentially the same. The homeomorphism, ϕ\phi assigns to any point in U a point in Rn so assigns to any point in U n numbers, the "parameterization" or "coordinates" for that point.

So, in the example of a 2-sphere, $S^{2}\subset\mathbb{R}^{3}$, am I correct in saying that the homeomorphism, $\phi :U\subset S^{2}\rightarrow V\subset\mathbb{R}^{2}$ assigns coordinates $(\theta , \phi)\in\mathbb{R}^{2}$ to a given point $p\in U$ and then the inverse map $\phi^{-1}:V\rightarrow U$ parameterizes that point (as a function of the coordinates $(\theta , \phi)$), such that $(\sin(\theta)\cos(\phi),\sin(\theta)\sin(\phi),\cos(\theta))\in S^{2}\subset\mathbb{R}^{3}$ describes the point $p\in U\subset S^{2}$? In other words, each value of coordinates $(\theta , \phi)$ labels a unique point $p\in U$ such that we can unambiguously describe that point, directly on the sphere, in terms of the coordinates assigned to the point.

Would it be correct to say that, in general in mathematics, a parameterization of an object is a way of describing points on that object in terms of certain attributes of the object (such as angles of longitude and latitude) in a well-defined manner, such that each specific set of values for the parameters determine a unique point on the object?

It implies that one can use Cartesian coordinates, but not that one can only use Cartesian coordinates. An obvious example is R3 itself where you can use Cartesian coordinates, spherical coordinates, cylindrical coordinates, etc.

Am I correct in thinking that the Cartesian coordinate system is an identity map on $\mathbb{R}^{n}$, such that the parameterization of each point in $\mathbb{R}^{n}$ is equal to its coordinate values? If so, it makes sense to me why one cannot use Cartesian coordinates on more general manifolds, as this requires them to be isometric to $\mathbb{R}^{n}$.

 

[WWGD]

I assume you mean that a zero-curvature subspace S must be of the type x(s)=s+c ; c a constant Well, if you can parametrize S, unit speed , then you would have curature as the norm of ||x'(t)|| , so that if the curvature is zero, then, integrating, ||x'(s)||=0 , you end up getting x(s)=s+c.

And you need to be more specific if you want good answers. What do you mean by, e.g., "whatis the difference"? What kind of a difference are you looking for?

And if I understood correctly, then, yes, the chart map do give a local parametrization of embedded manifolds in $\mathbb R^n$, but the parametrization describes a whole chart/open set, not just a point.

 

["Don't panic!"]

I assume you mean that a zero-curvature subspace S must be of the type x(s)=s+c ; c a constant Well, if you can parametrize S, unit speed , then you would have curature as the norm of ||x'(t)|| , so that if the curvature is zero, then, integrating, ||x'(s)||=0 , you end up getting x(s)=s+c.

I was really trying to understand correctly the exact definition of a Cartesian coordinate system. Is it just the case where one assigns coordinates $(x^{1},\ldots ,x^{n})$ to a patch on a manifold such that the parametrization of each point is given by the map $(x^{1},\ldots ,x^{n})\mapsto (x^{1},\ldots ,x^{n})$?

And you need to be more specific if you want good answers. What do you mean by, e.g., "whatis the difference"? What kind of a difference are you looking for?

Is there any fundamental difference between the coordinates of a point and the parametrization of that point? Is it simply that the coordinates are labels that allow one to distinguish individual points in a patch on the manifold, and then we can use these to parametrize the patch such that each value of the coordinates describes a given point on the manifold (in terms of the parametrization). For example, if $(\theta , \phi)$ are coordinates for a hemisphere of $S^{2}$, then we can describe the points directly via the parametrization $(\sin(\theta)\cos(\phi),\sin(\theta)\sin(\phi),\cos(\theta))$, such that if $(\theta_{1} , \phi_{1})$ are the coordinates corresponding to a point $p_{1}$ then $(\sin(\theta_{1})\cos(\phi_{1}),\sin(\theta_{1})\sin(\phi_{1}),\cos(\theta_{1}))$ is the unique (in terms of this particular coordinate chart) description of this point. Likewise, if $(\theta_{2} , \phi_{2})$ are the coordinates corresponding to another point $p_{2}$ then $(\sin(\theta_{2})\cos(\phi_{2}),\sin(\theta_{2})\sin(\phi_{2}),\cos(\theta_{2}))$ is the unique (in terms of this particular coordinate chart) description of this point?

 

[lavinia]

Here is an example. Each point on the Earth is assigned a longitude and a latitude. This a point in the Euclidean plane. Over the sphere minus the north and south poles and one half great circle connecting them, longitude and latitude coordinates are a homeomorphism. Conversely give an longitude and latitude, one can retrieve the point on the Earth that they parametrize.

In general, coordinates do not preserve shape. They are required only to be homeomorphisms (or diffeomorphisms in the case of differentiable manifolds). The manifold need not have a Riemannian metric. The idea of coordinate charts and parameterizations is independent of geometry. However, if the manifold has a metric, then one can ask whether there are coordinate charts that preserve the metric. or some aspect of it If the manifold is flat, that is it has a Riemannian metric whose curvature tensor is identically zero, then coordinate charts can be chosen to be local isomteries with open subsets of Euclidean space( with the standard Euclidean metric).

Globally, a flat manifold need not be diffeomorphic to a subset of Euclidean space of the same dimension as WWGD has illustrated with the cylinder. Other examples of flat surfaces are the Mobius band made out of a piece of paper and the helicoid. The torus and the Klein bottle can also be given metrics of zero curvature.

A good exercise is to show that the map $$(x,y) -> \sqrt 2 (cos(2πx),sin(2πx),cos(2πy),sin(2πy))$$ is a local isometry of the Euclidean plane onto a flat torus. By appropriately restricting x an y one can make this map into a collection of parametrizations of open neighborhoods on this flat torus.

One can also think of the torus as the quotient space of the Euclidean plane by the action of a lattice, for instance the lattice of points in the plane with integer coordinates. The quotient space is diffemorphic to a torus and the projection map defines a collection of local parameterizations. Additionally, the quotient "inherits" the flat Riemannian metric from the plane so the projection map is again a local isometry.

Note that in these two cases of the flat torus, coordinate charts are not considered, only parameterizations. One could of course, invert these parameterizations to obtain coordinate charts.

 

["Don't panic!"]

Conversely give an longitude and latitude, one can retrieve the point on the Earth that they parametrize.

So is the point then that the coordinates are labels that allow one to distinguish individual points in a patch on the manifold, and then we can use these to parametrize the patch such that each value of the coordinates describes a given point on the manifold (in terms of the parametrization). In practice then is the inverse map used more often as one can parametrize a given patch by choosing a particular set of coordinates? (When the notion of a coordinate chart is introduced is it mainly to state the fact that one can associate each point in a patch with a unique coordinate in Euclidean space, such that we can use these coordinate maps to parametrize a given patch on the manifold?)

How should one define Cartesian coordinates then? Are they essentially identity maps from $\mathbb{R}^{n}$ to $\mathbb{R}^{n}$, such that the parametrization of a point is simply its coordinates in $\mathbb{R}^{n}$?

 

[lavinia]

So is the point then that the coordinates are labels that allow one to distinguish individual points in a patch on the manifold, and then we can use these to parametrize the patch such that each value of the coordinates describes a given point on the manifold (in terms of the parametrization). In practice then is the inverse map used more often as one can parametrize a given patch by choosing a particular set of coordinates? (When the notion of a coordinate chart is introduced is it mainly to state the fact that one can associate each point in a patch with a unique coordinate in Euclidean space, such that we can use these coordinate maps to parametrize a given patch on the manifold?)

I don't think there is a preference for charts over parameterizations.

How should one define Cartesian coordinates then? Are they essentially identity maps from $\mathbb{R}^{n}$ to $\mathbb{R}^{n}$, such that the parametrization of a point is simply its coordinates in $\mathbb{R}^{n}$?

Standard Cartesian coordinates are just the identity maps as you have said but of course there are other possibilities. In fact any homeomorphism/diffeomorphism of a set in Euclidean space into another is also a coordinate chart. Euclidean space is just another manifold.

 

["Don't panic!"]

I don't think there is a preference for charts over parameterizations.

So is the point that the coordinate map $\phi$ assigns coordinates in $\mathbb{R}^{n}$ to each point in a patch on the manifold, and the inverse map $\phi^{-1}$ parametrizes each point in the patch in terms of its coordinates in $\mathbb{R}^{n}$, such that $\phi^{-1}(x^{1},\ldots ,x^{n})=(u^{1},\dots ,u^{n}) \in M$, where each $u^{i}$ is a function of coordinates $(x^{1},\ldots ,x^{n})\in\mathbb{R}^{n}$?

 

[micromass]

So is the point that the coordinate map $\phi$ assigns coordinates in $\mathbb{R}^{n}$ to each point in a patch on the manifold, and the inverse map $\phi^{-1}$ parametrizes each point in the patch in terms of its coordinates in $\mathbb{R}^{n}$, such that $\phi^{-1}(x^{1},\ldots ,x^{n})=(u^{1},\dots ,u^{n}) \in M$, where each $u^{i}$ is a function of coordinates $(x^{1},\ldots ,x^{n})\in\mathbb{R}^{n}$?

Why would points in $M$ have the form $(u_1,...,u_n)$?

 

["Don't panic!"]

Why would points in MM have the form (u1,...,un)(u_1,...,u_n)?

I guess not in general. I was trying to understand the general idea by generalising from the example of a 2-sphere $S^{2}\subset{R}^{3}$, in which a patch $U\subset S^{2}$ on can be assigned spherical polar coordinates $(\theta , \phi)$ such that $p\in U\subset M\mapsto (\theta , \phi)\in V\subset\mathbb{R}^{2}$ (where $V=\lbrace (\theta , \phi)\vert \theta\in (0, \pi),\;\phi\in (0,2\pi)\rbrace$). Each point in this patch can then be parametrized in terms of its coordinates in $\mathbb{R}^{2}$ in the following manner, $(\theta , \phi)\in V\subset\mathbb{R}^{2}\mapsto (\sin(\theta)\cos(\phi),\sin(\theta)\sin(\phi),\cos(\theta))=p\in U\subset S^{2}$?!