# Charts on Topological Manifolds - Simple Notational Issue

1. Feb 20, 2016

### Math Amateur

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) ...

My question regarding this definition is as follows:

What is the meaning of $M$ and how does it differ from $M^m$?

Surely the relationship between $M$ and $M^m$ is not the same as the relationship between $R$ and $R^m$ ... ???

I am not even sure what $M$ 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 ... ...

#### Attached Files:

File size:
45.4 KB
Views:
164
File size:
100.6 KB
Views:
190
File size:
106.7 KB
Views:
179
• ###### Barden & Thomas - 3 - Ch 1 - page 3 .png
File size:
115.9 KB
Views:
147
2. Feb 20, 2016

### Staff: Mentor

$M^m$ 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.

3. Feb 21, 2016

### Math Amateur

Thanks fresh_42 ... appreciate the help ...

Peter