- #1

Arkuski

- 40

- 0

**Problem Statement**

Suppose [itex]M_1,...,M_k[/itex] are smooth manifolds and [itex]N[/itex] is a smooth manifold with boundary. Then [itex]M_1×..×M_k×N[/itex] is a smooth manifold with a boundary.

**Attempt**

Since [itex]M_1,...,M_k[/itex] are smooth manifolds, we are allowed to use the theorem that states that [itex]M_1×...×M_k[/itex]is smooth with charts [itex](U_1×...×U_k,\phi _1×...×\phi _k)[/itex].

I get the idea that given the smooth manifold $N$ with coordinate chart [itex](U,\psi _i)[/itex] where [itex]\psi _i:N\rightarrow H^n[/itex] ([itex]H^n[/itex] is the half plane of dimension [itex]n[/itex]), we show that [itex]H^n[/itex] restricts [itex]x_i[/itex] 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.