Diffeomorphism From Tangent Bundle to Product

  • Thread starter Arkuski
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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.
 

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  • #2
fzero
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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.)
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.

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.
It is impossible to cover ##S^1## with a single chart, since the endpoints of the line would map to the same point.
 
  • #3
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Have ##M## and ##N## been defined?
My mistake, I meant [itex]S^1×ℝ[/itex] instead of [itex]M×N[/itex]
 
  • #4
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It is impossible to cover ##S^1## with a single chart, since the endpoints of the line would map to the same point.
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.

My mistake, I meant [itex]S^1×ℝ[/itex] instead of [itex]M×N[/itex]
You mean "instead of ##TM\times TN##"?
 
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  • #5
fzero
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My mistake, I meant [itex]S^1×ℝ[/itex] instead of [itex]M×N[/itex]
You mean "instead of ##TM\times TN##"?
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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