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Math Help - Tangent Bundle of S^1

  1. #1
    Senior Member slevvio's Avatar
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    Tangent Bundle of S^1

    I was asked to show that TS^1 is diffeomorphic to S^1 \times \mathbb{R}.

    But why can I not just use the map TS^1 \rightarrow S^1 \times \mathbb{R} given by v \in T_p S^1 \mapsto (p, v), since each T_p S^1 = \mathbb{R}?

    I know this cannot be the correct map, but I am not sure why. Any help would be appreciated.
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  2. #2
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    Re: Tangent Bundle of S^1

    If you can do that way you can do the same to TS^2, which is not a trivial bundle.
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  3. #3
    Senior Member slevvio's Avatar
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    Re: Tangent Bundle of S^1

    I know it doesn't work but I can't see why, every way I look at it I seem to get a diffeomorphism if I write that map down for TM \rightarrow M \times \mathbb{R}^{\operatorname{dim}M} for any manifold
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  4. #4
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    Re: Tangent Bundle of S^1

    TM cannot always be covered by a single chart. Your approach only proved that in that single chart the two are diffeomorphism. This is obvious since the bundle is patched together in such a way. Your argument doensn't state anything globally, which is needed.
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  5. #5
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    Re: Tangent Bundle of S^1

    To prove two space are diffemorphic you need to prove 1) they're homeomorphic 2) both the map and inverse map are smooth, which can be done locally like your approach. So you just missed 1).
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  6. #6
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    Re: Tangent Bundle of S^1

    You even didn't define a map well since you only defined in one chart. You need to patch the charts to get a well-defined map. That is, the definition coincides on overlapped charts.
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  7. #7
    Senior Member slevvio's Avatar
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    Re: Tangent Bundle of S^1

    I think I see what you are saying, thank you
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