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.

Examples

• Euclidean space, ${ℝ}^n$ with a single chart ($O={ℝ}^n,\psi =$ identity map) is a trivial example of a manifold.
• 2-sphere $S^2=\{(x,y,z)\in {ℝ}^3|x^2+y^2+z^2=1\}$.
• ...