General Relativity/Differentiable manifolds
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< General Relativity(Redirected from General relativity/Differentiable manifolds)
A smooth n-dimensional manifold Mn is a set together with a collection of subsets {Oα} with the following properties:
- Each
lies in at least one Oα, that is
. - For each α, there is a bijection
, where Uα 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 Oα of the charts are called coordinate regions.
[edit] Examples
- Euclidean space,
with a single chart (
identity map) is a trivial example of a manifold. - 2-sphere
.
- Notice that S2 is not an open subset of
. The identity map on
restricted to S2 does not satisfy the requirements of a chart since its range is not open in 
- The usual spherical coordinates map S2 to a region in
, but again the range is not open in
Instead, one can define two charts each defined on a subset of S2 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 S2. With these two charts, S2 becomes a 2-dimensional manifold. It can be shown that no single chart can possibly cover all of S2 if the topology of S2 is to be the usual one.
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lies in at least one
.
, where 
is non-empty, then the map
is
.
. The identity map on 
, but again the range is not open in
Instead, one can define two charts each defined on a subset of