General Relativity/Differentiable manifolds

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<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[edit]

  • 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.