Charts on Topological Manifolds - Simple Notational Issue

In summary, the conversation is about a question regarding the definition of a chart in "An Introduction to Differential Topology" by Dennis Barden and Charles Thomas. The question is about the meaning of M and how it differs from M^m, and whether the relationship between M and M^m is the same as the relationship between R and R^m. The response clarifies that M and M^m are the same, with M^m just being a reminder of the manifold's dimension. The author of the book uses this notation to emphasize the dimension of the coordinates.
  • #1
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I am reading "An Introduction to Differential Topology" by Dennis Barden and Charles Thomas ...

I am focussed on Chapter 1: Differential Manifolds and Differentiable Maps ...

I need some help and clarification on an apparently simple notational issue regarding the definition of a chart (Definition 1.1.3) ...

Definition 1.1.3 reads as follows:
?temp_hash=df27f89f2e21dffb8b683d0466b294ab.png
My question regarding this definition is as follows:

What is the meaning of [itex]M[/itex] and how does it differ from [itex]M^m[/itex]?

Surely the relationship between [itex]M[/itex] and [itex]M^m[/itex] is not the same as the relationship between [itex]R[/itex] and [itex]R^m[/itex] ... ?

I am not even sure what [itex]M[/itex] is ... ?

Can someone clarify the above issue for me ...?

Hope someone can help ...

Peter===========================================================

So that readers can understand the context and notation of Barden and Thomas, I am providing the pages of the text leading up to and including the definition referred to above ... ... as follows ... ...
?temp_hash=df27f89f2e21dffb8b683d0466b294ab.png

?temp_hash=df27f89f2e21dffb8b683d0466b294ab.png

?temp_hash=df27f89f2e21dffb8b683d0466b294ab.png
 

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    Barden & Thomas - 1 - Ch 1 - page 1.png
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  • #2
##M^m## is just a reminder that the manifold ##M## is of ##m## dimensions. So ##M^m = M##.
I have never seen such a notation but it's clear from definition 1.1.1. It only symbolizes that the (Euclidean) coordinates are in ##ℝ^m##, i.e. there are ##m## coordinates. I guess the author drops the ##m## in ##M^m## and sticks with ##M## when he doesn't need to emphasize the dimension.
 
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  • #3
Thanks fresh_42 ... appreciate the help ...

Peter
 

1. What are topological manifolds?

Topological manifolds are mathematical spaces that are locally similar to Euclidean spaces, but may have more complicated global structure. They are often used to model geometric objects in physics, engineering, and other branches of mathematics.

2. What is the purpose of charts on topological manifolds?

Charts on topological manifolds are used to define a coordinate system on the manifold, allowing for the representation and study of geometric objects in a more familiar framework. They also allow for the ability to perform calculations and make comparisons between different points on the manifold.

3. Why is there a notational issue with charts on topological manifolds?

The notational issue arises because different authors and fields of mathematics may use different symbols and conventions to represent charts on topological manifolds. This can cause confusion and make it difficult to communicate ideas between different sources.

4. How can the notational issue with charts on topological manifolds be addressed?

To address the notational issue, it is important for authors to clearly define their notation and conventions when discussing charts on topological manifolds. It can also be helpful for readers to have a basic understanding of the different notations used in different fields to better interpret and compare different sources.

5. Are there any standard notations for charts on topological manifolds?

While there is no universally accepted standard notation for charts on topological manifolds, there are some commonly used symbols and conventions. For example, many authors use Greek letters such as φ and ψ to represent charts, and use the notation U and V for open sets in the manifold. However, it is important to always refer to the specific notation used in a particular source to avoid confusion.

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