Product of Smooth Manifolds and Boundaries

[Arkuski]

Sorry guys, I have some differential topology homework, and I may be asking a lot of questions in the next few days.

Problem Statement
Suppose M_1,...,M_k are smooth manifolds and N is a smooth manifold with boundary. Then M_1×..×M_k×N is a smooth manifold with a boundary.

Attempt
Since M_1,...,M_k are smooth manifolds, we are allowed to use the theorem that states that M_1×...×M_kis smooth with charts (U_1×...×U_k,\phi _1×...×\phi _k).

I get the idea that given the smooth manifold $N$ with coordinate chart (U,\psi _i) where \psi _i:N\rightarrow H^n (H^n is the half plane of dimension n), we show that H^n restricts x_i to non-negative values, and then the entire product in consideration will only have one coordinate restricted to non-negative values, signifying the presence of a boundary. I'm just really confused about how to put it into words.

 

[pasmith]

Sorry guys, I have some differential topology homework, and I may be asking a lot of questions in the next few days.

Problem Statement
Suppose M_1,...,M_k are smooth manifolds and N is a smooth manifold with boundary. Then M_1×..×M_k×N is a smooth manifold with a boundary.

Attempt
Since M_1,...,M_k are smooth manifolds, we are allowed to use the theorem that states that M_1×...×M_kis smooth with charts (U_1×...×U_k,\phi _1×...×\phi _k).

I get the idea that given the smooth manifold $N$ with coordinate chart (U,\psi _i) where \psi _i:N\rightarrow H^n (H^n is the half plane of dimension n), we show that H^n restricts x_i to non-negative values, and then the entire product in consideration will only have one coordinate restricted to non-negative values, signifying the presence of a boundary. I'm just really confused about how to put it into words.

The theorem you quoted shows that M_1 \times \cdots \times M_k is a smooth manifold, so it's sufficient to prove the result for M \times N, where M is an arbitrary smooth manifold.

Take an arbitrary point (x,y) \in M \times N. Then take a chart on M centered at x and a chart on N centered at y, work out what the codomain of the product chart is, and confirm that the chart is a homeomorphism.

Then you'll need to show that all product charts are smoothly compatible.