The tangent bundle $TS^n$ is diffeomorphic the $S^n \times \mathbb{R}^n$?
Comments: I was able to show that $TS^1$ is diffeomorphic the $S^1 \times \mathbb{R}$. For the case $n=2$, I believe it is not true, but I do not know how to justify it.
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what was you diffeomorphic map for the case $n=1$? – Arian Apr 27 '18 at 15:40
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Hint: hairy ball theorem. – Wojowu Apr 27 '18 at 15:40
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@Wojowu We posted at the exact same time. – Jacky Chong Apr 27 '18 at 15:41
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@Arian the map: $(p, \lambda \frac{d}{dt}(p)) \mapsto (p,\lambda)$ – Croos Apr 27 '18 at 15:46
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1@Croos: Although it's not the subject of your question, showing $TS^1$ is diffeomorphic to $S^1\times\mathbb{R}$ takes a bit more work than the map you posted in the comment above: that only works locally. – Steve D Apr 27 '18 at 16:21
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@Croos Just so you know, this story about for which $n$ we have $TS^n=S^n \times \mathbb{R^n}$ will get you very quickly into deep mathematics. Persue it! – Nick A. Apr 27 '18 at 17:08
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This seems to be a duplicate of https://math.stackexchange.com/questions/1323740/are-t-mathbbs-2-and-mathbbs-2-times-mathbbr2-different if one only cares about the case of even $n$. – Jason DeVito - on hiatus Apr 29 '18 at 02:09
