- #1

"Don't panic!"

- 601

- 7

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

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 [itex]S^{2}\subset\mathbb{R}^{3}[/itex] (considering it as a subset of [itex]\mathbb{R}^{3}[/itex], i.e. essentially embedded in [itex]\mathbb{R}^{3}[/itex]), then is the 2-tuple [itex](\theta , \phi)\in\mathbb{R}^{2}[/itex] the coordinates of a point on the manifold (with the mapping defined by [itex]p\mapsto (\theta , \phi)[/itex]) and its corresponding local parametrization on the manifold, [itex](\sin (\theta)\cos (\phi), \sin (\theta)\sin (\phi), \cos (\theta))\in S^{2}\subset\mathbb{R}^{3}[/itex] (with the inverse mapping defined by [itex](\theta , \phi)\mapsto (\sin (\theta)\cos (\phi), \sin (\theta)\sin (\phi), \cos (\theta))[/itex]) ?

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 [itex](M,g)[/itex] is locally flat (i.e. has vanishing local curvature) then there will be an isometry [itex]\psi[/itex] to an open set in [itex](\mathbb{R}^{n},\bar{g})[/itex] (where [itex]g[/itex] is the metric defined on the [itex]n[/itex]-dimensional manifold [itex]M[/itex], and [itex]\bar{g}[/itex] is the Euclidean metric defined on [itex]\mathbb{R}^{n}[/itex]).

*local parametrization*to a point in a given patch on a manifold. By this is it meant that, given an [itex]n[/itex]-dimensional manifold [itex]M[/itex] and a homeomorphism [itex]\phi :U\subset M\rightarrow V\subset\mathbb{R}^{n}[/itex] from a patch on the manifold [itex]U\subset M[/itex], then we can parametrize a point [itex]p\in U[/itex] via the inverse map [itex]\phi^{-1}:V\subset\mathbb{R}^{n}\rightarrow U\subset M[/itex]. More explicitly, if [itex]\phi (p)=(x^{1},\ldots ,x^{n})[/itex] are the coordinates of [itex]p[/itex] in [itex]\mathbb{R}^{n}[/itex], then [tex]p=(\phi^{-1}\circ\phi )(p)=\phi^{-1}(\phi(p))=\phi^{-1}(x^{1},\ldots ,x^{n})=(u^{1},\ldots ,u^{n})[/tex] where [itex](u^{1},\ldots ,u^{n})[/itex] is the local parametrization of [itex]p[/itex] on [itex]M[/itex], with [itex]u^{i}=u^{i}(x^{1},\ldots ,x^{n})[/itex] are functions whose domain is [itex]\mathbb{R}^{n}[/itex].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 [itex]S^{2}\subset\mathbb{R}^{3}[/itex] (considering it as a subset of [itex]\mathbb{R}^{3}[/itex], i.e. essentially embedded in [itex]\mathbb{R}^{3}[/itex]), then is the 2-tuple [itex](\theta , \phi)\in\mathbb{R}^{2}[/itex] the coordinates of a point on the manifold (with the mapping defined by [itex]p\mapsto (\theta , \phi)[/itex]) and its corresponding local parametrization on the manifold, [itex](\sin (\theta)\cos (\phi), \sin (\theta)\sin (\phi), \cos (\theta))\in S^{2}\subset\mathbb{R}^{3}[/itex] (with the inverse mapping defined by [itex](\theta , \phi)\mapsto (\sin (\theta)\cos (\phi), \sin (\theta)\sin (\phi), \cos (\theta))[/itex]) ?

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 [itex](M,g)[/itex] is locally flat (i.e. has vanishing local curvature) then there will be an isometry [itex]\psi[/itex] to an open set in [itex](\mathbb{R}^{n},\bar{g})[/itex] (where [itex]g[/itex] is the metric defined on the [itex]n[/itex]-dimensional manifold [itex]M[/itex], and [itex]\bar{g}[/itex] is the Euclidean metric defined on [itex]\mathbb{R}^{n}[/itex]).

Last edited: