Product of Smooth Manifolds and Boundaries

In summary, the discussion involved using the theorem stating that M_1 \times \cdots \times M_k is a smooth manifold, and then applying this to the specific case of M \times N. The approach is to take an arbitrary point and use charts to show that the product chart is a homeomorphism. The final step is to prove that all product charts are smoothly compatible.
  • #1
Arkuski
40
0
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.
 
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  • #2
Arkuski said:
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.
 

1. What is a product of smooth manifolds and boundaries?

A product of smooth manifolds and boundaries is a mathematical concept that combines two or more smooth manifolds, which are mathematical spaces that locally look like Euclidean space, along with their respective boundaries, which are lower-dimensional subsets of the manifolds. This allows for the study of more complex spaces that cannot be described by a single manifold.

2. How are smooth manifolds and boundaries related?

Smooth manifolds and boundaries are closely related as they both describe geometric spaces. A smooth manifold is a space that is locally Euclidean, while a boundary is a lower-dimensional subset of that space. The product of these two concepts allows for the study of spaces that have both smooth and non-smooth components.

3. What are some applications of products of smooth manifolds and boundaries?

The product of smooth manifolds and boundaries has many applications in mathematics and physics. It is used to study complex spaces such as the moduli space of Riemann surfaces and to define the concept of a vector bundle. It is also applied in differential geometry, topology, and algebraic geometry.

4. How is the product of smooth manifolds and boundaries different from a Cartesian product?

The product of smooth manifolds and boundaries is different from a Cartesian product in that it combines not just the underlying sets of the manifolds, but also their smooth structures and boundaries. This allows for a more nuanced and complex space to be studied, as opposed to just a direct combination of two simpler spaces.

5. What are some properties of products of smooth manifolds and boundaries?

Products of smooth manifolds and boundaries possess many important properties such as smoothness, Hausdorffness, and paracompactness. They also have a natural topology and can be equipped with a tangent bundle, which allows for the study of differential equations on these spaces. Additionally, the product of smooth manifolds and boundaries has a well-defined product structure that allows for operations such as multiplication and addition.

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