Smooth Charts on immersion image.

In summary, the topology on $N'$ is induced from $N$ and the smooth structure is defined by using local diffeomorphisms.
  • #1
WWGD
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Hi, everyone:

I have been reading Boothby's intro to diff. manifolds, and in def. 4.3,
talking about 1-1 immersions F:N->M , n and m-mfld. and M an m-mfld.
That:

"...F establishes a 1-1 correspondence between N and the image N'=F(N)
of M. If we use this correspondence to give N' a topology and a C^oo
structure, then N' will be called a submanifold, and F:N->N' a diffeomorphism".

I am just wondering how this topology and C^oo structure are defined.
I imagine we are using pullbacks here, tho, since F is not necessarily
a diffeom. , we cannot pullback by F, except maybe locally, using
the inverse fn. theorem.
And, re the topology on F'(N) , we cannot use subspace, since
F may not be an embedding. Maybe we are using quotient topology,
but it does not seem clear.


Anyone know how the topology and C^oo structure are defined
in N'=F(N)?


I would appreciate any help.
 
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  • #2
Thanks.When defining the topology on $N'$, we use the fact that $F$ is an immersion to identify $N'$ with an open subset of $N$. Thus, the topology on $N'$ is the subspace topology induced from the ambient space $N$. For the smooth structure on $N'$, we can use the fact that $F$ is a local diffeomorphism. Because of this, for each point $p \in N' = F(N)$, there is an open neighborhood $U_p \subset N'$ and an open neighborhood $V_p \subset N$ such that $F : V_p \to U_p$ is a diffeomorphism. Thus, we can define the smooth structure on $N'$ by giving each $p \in N'$ a chart $(U_p, \phi_p)$ where $\phi_p = F^{-1} |_{U_p}$. We then check that these charts are compatible with each other (i.e., their transition functions are smooth).
 
  • #3
Thanks!

Hello,

Thank you for sharing your thoughts on Boothby's intro to differential manifolds. To answer your question, the topology and C^oo structure on the image N'=F(N) are defined using the pullback of the charts on N. Since F is a 1-1 immersion, it preserves the smooth structure and topology of N. This means that for each chart (U,ψ) on N, we can define a chart (F(U),ψ∘F^-1) on N'=F(N). This gives N' the same smooth structure and topology as N, making it a submanifold of M.

As for the C^oo structure, it is defined using the pullback of the smooth functions on M. Since F is a 1-1 immersion, it pulls back smooth functions on M to smooth functions on N. This means that for any smooth function f on M, we can define a smooth function F*(f) on N. This allows us to define a C^oo structure on N'=F(N) by taking the collection of all smooth functions F*(f) as a basis.

I hope this helps clarify how the topology and C^oo structure are defined on the immersion image. Let me know if you have any further questions. Happy studying!
 

1. What is a smooth chart on immersion image?

A smooth chart on immersion image refers to a graphical representation of data on a two-dimensional plane, where the data points are connected by a smooth line. This type of chart is often used to visualize trends and patterns in data over time.

2. How is a smooth chart on immersion image created?

A smooth chart on immersion image is typically created using a software program or online tool that allows the user to input their data and customize the chart's appearance. The program then uses algorithms to generate a smooth line that best fits the data points, creating the chart.

3. What are the advantages of using a smooth chart on immersion image?

Smooth charts on immersion image can make it easier to identify trends and patterns in data, as the smooth line can help to eliminate noise and fluctuations. They also provide a visually appealing way to present data and can make it easier for viewers to understand the information being conveyed.

4. Are there any limitations to using smooth charts on immersion image?

One limitation of using smooth charts on immersion image is that they may oversimplify the data, making it difficult to see the exact values for each data point. They also rely on the accuracy of the data input and may not accurately represent extreme or outlier data points.

5. How can I choose the best type of smooth chart on immersion image for my data?

The best type of smooth chart on immersion image will depend on the type of data you have and the story you want to tell with the data. Some common types of smooth charts include line charts, scatter plots, and area charts. It may be helpful to experiment with different types of charts to see which one best represents your data and conveys your message effectively.

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