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 ${\psi }_{\alpha }:O_{\alpha }⟶U_{\alpha }$, where $U_{\alpha }$ is an open subset of ${ℝ}^n$
3. If $O_{\alpha }\cap O_{\beta }$ is non-empty, then the map ${\psi }_{\alpha }\circ {\psi }_{\beta }^{-1}:{\psi }_{\beta }[O_{\alpha }\cap O_{\beta }]⟶{\psi }_{\alpha }[O_{\alpha }\cap O_{\beta }]$ 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, ${ℝ}^n$ with a single chart ($O={ℝ}^n,\psi =$ identity map) is a trivial example of a manifold.
• 2-sphere $S^2=\{(x,y,z)\in {ℝ}^3|x^2+y^2+z^2=1\}$.

Notice that $S^2$ is not an open subset of ${ℝ}^3$. The identity map on ${ℝ}^3$ restricted to $S^2$ does not satisfy the requirements of a chart since its range is not open in ${ℝ}^3.$

The usual spherical coordinates map $S^2$ to a region in ${ℝ}^2$, but again the range is not open in ${ℝ}^2.$ 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.