# Tangent Bundle of S^1

• Nov 12th 2011, 02:06 PM
slevvio
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.
• Nov 13th 2011, 05:59 AM
xxp9
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.
• Nov 13th 2011, 07:56 AM
slevvio
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 :(
• Nov 13th 2011, 08:19 AM
xxp9
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.
• Nov 13th 2011, 08:21 AM
xxp9
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).
• Nov 13th 2011, 08:36 AM
xxp9
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.
• Nov 14th 2011, 06:22 AM
slevvio
Re: Tangent Bundle of S^1
I think I see what you are saying, thank you