Manifold & Manifold with Boundary
An
-dimensional manifold, or -manifold (without boundary) for short, is a Hausdorff and second-countable topological space with the property that each point has a neighbourhood that is homeomorphic to an open subset of . A manifold with boundary is a Hausdorff and second-countable topological space such that each point has a neighbourhood homeomorphic to an open subset of the closed half-space . The boundary of is the set of points that correspond to points in with . A closed manifold is a compact manifold with no boundary.
Clearly, on a manifold, there might be different collections of homeomorphisms that makes it locally Euclidean. So we introduce:
Chart and Atlas
A chart for a manifold
is a homeomorphism where is open in and is open in . A collection of charts is called an atlas for if . A manifold equipped with an atlas is called a topological manifold.
e.g.
- Consider the set
with the usual topology. and are both differentiable atlases. They are not compatible since the union is not differentiable. - The sphere
is a closed subset of Euclidean space, thus the topological requirements are satisfied. Define the following two maps: as follows: We write as where and , we take and . Then we have
Compatible Charts & Smooth Atlas
Two charts
and are compatible if is a diffeomorphism. Let be an -dimensional manifold. An atlas for is differentiable or smooth if for every and in , the map and are compatible.
Compatible Atlases & Maximal Atlas
Two differentiable atlases
and are compatible if their union is also a differentiable atlas. A smooth atlas is called maximal if it is not contained in any other atlas. In other words, a maximal atlas is the one that contains all charts of that are compatible with .
Proposition
Two differentiable atlases
and are compatible if and only if for every chart in and in , both and are smooth.
Smooth Manifold
A
-structure on a manifold is an equivalence class of differentiable atlases, where two atlases are deemed equivalent if they are * -compatible. A smooth manifold is a manifold together with a smooth ( ) structure on .
e.g.
- A given manifold can carry many different atlases: For example, if
we can take two charts and . Then both and are smooth atlases for , they define different smooth atlases, while they are compatible. - The 4-manifold
is a topological manifold, but not smooth.
In fact, dimension 4 is particularly special and exotic in the theory of smooth manifolds, there are smooth 4-manifolds which are homeomorphic but not diffeomorphic. We have the following theorem:
Theorem
for all has a unique smooth structure. has uncountably many smooth structures.
Submanifolds
Submanifold of
A subset
is a -dimensional submanifold of , if for every there exists an open set containing and a diffeomorphism such that , with . We call such a map a submanifold chart for at .
e.g.
Proposition
Any
-dimensional submanifold of is a smooth manifold.
There are several equivalent ways to define submanifolds:
Proposition
Let
be a subset of . The following are equivalent:
is a -dimensional submanifold of ; is locally the graph of a smooth function. That is, for every , there is a neighbourhood of in , a linear injection and a complementary linear injection ,
an open subset, and a smooth map such that
is locally the level set of a submersion. That is, for every , there is a neighbourhood of in and a submersion such that
is locally the image of an embedding. That is, for every , there exists a neighbourhood of in , an open set , and a smooth embedding such that .
Embedded Submanifold
Suppose
is a smooth manifold with or without boundary. An embedded (or regular) submanifold of is a subset that is a manifold (without boundary) in the subspace topology, endowed with a smooth structure with respect to which the inclusion map is a smooth embedding.
Open Submanifold Lemma
An open subset of a manifold is a manifold of the same dimension.
Proof Let