Manifolds with Boundary
Ever since we started talking about manifolds, we’ve said that they locally “look like” the Euclidean space 
. We now need to be a little more flexible and let them “look like” the half-space 
.
Away from the subspace 
, 
is a regular 
-dimensional manifold — we can always find a small enough ball that stays away from the edge — but on the boundary subspace it’s a different story. Just like we wrote the boundary of a singular cubic chain, we write 
for this boundary. Any point 
that gets sent to 
by a coordinate map 
must be sent to by every coordinate map. Indeed, if 
is another coordinate map on the same patch 
around 
, then the transition function 
must be a homeomorphism from 
onto 
, and so it must send boundary points to boundary points. Thus we can define the boundary 
to be the collection of all these points.
Locally, is an 
-dimensional manifold. Indeed, if 
is a coordinate patch around a point 
then 
, and thus the preimage 
is an -dimensional coordinate patch around . Since every point is contained in such a patch, is indeed an -dimensional manifold.
As for smooth structures on 
and , we define them exactly as usual; real-valued functions on a patch of containing some boundary points are considered smooth if and only if the composition 
is smooth as a map from (a portion of) the half-space to 
. And such a function is smooth at a boundary point of the half-space if and only if it’s smooth in some neighborhood of the point, which extends — slightly — across the boundary.
Like this:
Like Loading...
September 13, 2011 -
Posted by John Armstrong |
Differential Topology, Topology
[...] we have a manifold with boundary , then at all the interior points it looks just like a regular manifold, and so the tangent space [...]
Pingback by The Tangent Space at the Boundary « The Unapologetic Mathematician | September 15, 2011 |
[...] take a manifold with boundary and give it an orientation. In particular, for each we can classify any ordered basis as either [...]
Pingback by Oriented Manifolds with Boundary « The Unapologetic Mathematician | September 16, 2011 |
[...] Now we come back to Stokes’ theorem, but in the context of manifolds with boundary. [...]
Pingback by Stokes’ Theorem on Manifolds « The Unapologetic Mathematician | September 16, 2011 |