General Relativity/Differentiable manifolds

A smooth $n$-dimensional manifold $\mathrm{M}^n$ is a set together with a collection of subsets $\{O_\alpha\}$ with the following properties:

1. Each $p\in\mathrm{M}$ lies in at least one $O_\alpha$, that is $\mathrm{M}=\cup_\alpha O_\alpha$.
2. For each $\alpha$, there is a bijection $\psi_\alpha:O_\alpha\longrightarrow U_\alpha$, where $U_\alpha$ is an open subset of $\mathbb{R}^n$
3. If $O_\alpha\cap O_\beta$ is non-empty, then the map $\psi_\alpha\circ\psi_\beta^{-1}:\psi_\beta[O_\alpha\cap O_\beta]\longrightarrow\psi_\alpha[O_\alpha\cap O_\beta]$ is smooth.

The bijections are called charts or coordinate systems. The collection of charts is called an atlas. The atlas induces a topology on M such that the charts are continuous. The domains $O_\alpha$ of the charts are called coordinate regions.

Examples

• Euclidean space, $\mathbb{R}^n$ with a single chart ($O=\mathbb{R}^n,\psi=$ identity map) is a trivial example of a manifold.
• 2-sphere $S^2 = \{ (x,y,z) \in \mathbb{R}^3 | x^2 + y^2 + z^2 = 1 \}$.
Notice that $S^2$ is not an open subset of $\mathbb{R}^3$. The identity map on $\mathbb{R}^3$ restricted to $S^2$ does not satisfy the requirements of a chart since its range is not open in $\mathbb{R}^3.$
The usual spherical coordinates map $S^2$ to a region in $\mathbb{R}^2$, but again the range is not open in $\mathbb{R}^2.$ Instead, one can define two charts each defined on a subset of $S^2$ that omits a half-circle. If these two half-circles do not intersect, the union of the domains of the two charts is all of $S^2$. With these two charts, $S^2$ becomes a 2-dimensional manifold. It can be shown that no single chart can possibly cover all of $S^2$ if the topology of $S^2$ is to be the usual one.
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