Chart, Parametrization and Atlas
- Differential Geometry 1.2
from 「Geometrical Methods of Mathematical Physics」: Bernard F. Schutz
Introduction
Topological manifold is one of topological spaces which is locally euclidean. Thus, we can think of coordinates which represents the neighborhood of a point of the topological manifold.
Atlas
Let’s think of $n$-dimensional topological manifold $M$ and its neighborhoods $U_\alpha$. Then following homeomorphisms $\varphi_\alpha$ exist, by its definition.
- $ \varphi_\alpha : U_\alpha \to \varphi_\alpha(U_\alpha) \sub \R^n $
Then we call $\varphi_\alpha$ a coordinate chart or a chart, and $\varphi_\alpha^{-1}$ a parametrization. These terms are very intuitive.
Also, we can think of a set which contains the pair of each neighborhood and chart in order to represent the whole manifold.
- $ \dps \mathcal{A} \coloneqq \Set{(U_\alpha,\varphi_\alpha) | \bigcup_{\alpha\in \Lambda} U_\alpha=M} $
The set $\mathcal{A}$ is called an atlas of the topological manifold $M$. If given the atlas, the topological manifold is called a ‘manifold’.
Transition map
Since points of different neighborhoods act on different charts, it is required to connect them in order to watch some global features(topology) of the manifold. It’s possible using transition maps.
Given $n$-dimensional manifold and its atlas $\mathcal{A}$, there’ll be neighborhoods which intersects. (This is assured by the Hausdorff condition of a topological manifold.)
- $ (U_\alpha,\varphi_\alpha),(U_\beta,\varphi_\beta)\in\mathcal{A},\; W=U_\alpha\cap U_\beta \not= \empty $
- $ \tau_{\alpha\beta} : \varphi_\alpha(W) \to \varphi_\beta(W),\; \tau_{\alpha\beta} \coloneqq \varphi_\beta \vert_W \circ \varphi_\alpha^{-1}\vert_W $
$\tau_{\alpha\beta}$ is a transition map, and it’s also a homeomorphism from a subset of $\R^n$ to a subset of $\R^n$. Therefore, it’s possible to talk about the continuity and differentiability of them.
An image describing transition map
