Thread: Manifolds Question View Single Post
2013-05-21, 13:43   #2
R.D. Silverman

Nov 2003

22×5×373 Posts

Quote:
 Originally Posted by jinydu Let M be a compact n-manifold with boundary. Show that there is an open set U containing the boundary that is diffeomorphic to (the boundary) x [0,1) -------- By definition of the boundary of a manifold, every point on the boundary has an open neighborhood that is diffeomorphic to R^{n-1} x [0,infinity), which of course I can shrink to R^{n-1} x [0,1). Also, by compactness, I can cover the boundary with finitely many such neighborhoods.
Looks good to me.

Quote:
 The hint here is that I'm supposed to use the theorem on existence of partitions of unity. Supposedly, that allows one to stitch together local information into global. Unfortunately, the book doesn't have any examples of that.
I don't understand the hint either.