Product of Smooth Manifolds and Boundaries

  • Thread starter Arkuski
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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 [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.
 

Answers and Replies

  • #2
pasmith
Homework Helper
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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 [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.

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

Take an arbitrary point [itex](x,y) \in M \times N[/itex]. Then take a chart on [itex]M[/itex] centered at [itex]x[/itex] and a chart on [itex]N[/itex] centered at [itex]y[/itex], 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.
 

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