Thread: Tangent Bundle of S^1

I was asked to show that is diffeomorphic to .

But why can I not just use the map given by , since each ?

I know this cannot be the correct map, but I am not sure why. Any help would be appreciated.

If you can do that way you can do the same to TS^2, which is not a trivial bundle.

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 for any manifold

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.

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).

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.

I think I see what you are saying, thank you