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Diffeomorphism From Tangent Bundle to Product

  1. Sep 29, 2013 #1
    Show that [itex]TS^1[/itex] is diffeomorphic to [itex]TM×TN[/itex].

    ([itex]TS^1[/itex] is the tangent bundle of the 1-sphere.)

    We can use the theorem stating the following.

    If [itex]M[/itex] is a smooth [itex]n[/itex]-manifold with or without boundary, and [itex]M[/itex] can be covered by a single smooth chart, then [itex]TM[/itex] is diffeomorphic to [itex]M×ℝ^n.[/itex]

    Clearly, I must be looking for a single smooth chart on [itex]S^1[/itex], but I am very uncertain on how to go about doing this. Any tips on managing the nuances are greatly appreciated.
     
  2. jcsd
  3. Sep 30, 2013 #2

    fzero

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    Have ##M## and ##N## been defined? ##TS^1## has a fairly simple geometric picture, but if we don't know what ##M## and ##N## are, we can't really help you.

    It is impossible to cover ##S^1## with a single chart, since the endpoints of the line would map to the same point.
     
  4. Sep 30, 2013 #3
    My mistake, I meant [itex]S^1×ℝ[/itex] instead of [itex]M×N[/itex]
     
  5. Sep 30, 2013 #4
    This. It's fairly well known that any atlas on ##S^1## must have at least two charts. The reason, as fzero mentions, is a chart must be equipped with a homeomorphism. For us to use a single chart here, if we start at a point, we must necessarily end on that point. Thus, the map from ##S^1## to ##\mathbb{R}## wouldn't even be single-valued, let alone a homeomorphism. Thus, we'd need two charts.

    You mean "instead of ##TM\times TN##"?
     
    Last edited: Sep 30, 2013
  6. Sep 30, 2013 #5

    fzero

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    OK, so let's assume that we have to show that ##TS^1## is diffeomorphic to ##S^1\times \mathbb{R}##. Since we know that we need at least two charts to cover ##S^1##, we know that we will need at least two charts to cover ##TS^1##.

    You should start by describing ##TS^1## at a point ##p## of ##S^1##. Then pick a covering set of charts for ##S^1## and describe the tangent space on each chart. Ultimately you will want to show that there is a basis for smooth vector fields that is valid at every point of ##S^1##.
     
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