General Relativity/Differentiable manifolds

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<General Relativity

A smooth n-dimensional manifold Mn is a set together with a collection of subsets {Oα} with the following properties:

  1. Each p\in\mathrm{M} lies in at least one Oα, that is \mathrm{M}=\cup_\alpha O_\alpha.
  2. For each α, there is a bijection \psi_\alpha:O_\alpha\longrightarrow U_\alpha, where Uα 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.

[edit] 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 \}.
  • ...
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