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Charts on Topological Manifolds - Simple Notational Issue

  1. Feb 20, 2016 #1
    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
     

    Attached Files:

  2. jcsd
  3. Feb 20, 2016 #2

    fresh_42

    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.
     
  4. Feb 21, 2016 #3
    Thanks fresh_42 ... appreciate the help ...

    Peter
     
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