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 $M_1×...×M_k$is 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 $M_1×...×M_k$is 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.