General Relativity/Differentiable manifolds
A smooth -dimensional manifold is a set together with a collection of subsets with the following properties:
- Each lies in at least one , that is .
- For each , there is a bijection , where is an open subset of
- If is non-empty, then the map 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 of the charts are called coordinate regions.
- Euclidean space, with a single chart ( identity map) is a trivial example of a manifold.
- 2-sphere .
- Notice that is not an open subset of . The identity map on restricted to does not satisfy the requirements of a chart since its range is not open in
- The usual spherical coordinates map to a region in , but again the range is not open in Instead, one can define two charts each defined on a subset of 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 . With these two charts, becomes a 2-dimensional manifold. It can be shown that no single chart can possibly cover all of if the topology of is to be the usual one.