General Relativity/Differentiable manifolds

<General Relativity

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.