For what values of n, not equal to 1,3,7, TS^n is diffeomorphic to an open subsets of R^{2n}?
Dear Joseph
Thank you very much for your answer.
I think there is no contradiction.
In fact TS^n is a parallelizable manifold, for all n.
Because we need to prove that TTS^n is a trivial bundle over TS^n. But the later can deformation retracts to S^n. On the other hand the restriction of the second tangent bundle to the (zero section of TS^n)~S^n is isomorphic to TS^n + TS^n. The later is trivial bundle since the complexification of TS^n is a trivial bundle, the later statement is written iwithout proof n a very old paper by Swan.
For a related discussion please see
https://mathoverflow.net/questions/151676/totally-non-parallelizable-manifold
Thank you for your attention to my question.
Yuri Ximenes Martins
Hello Yuri,I really appreciate your attention to my question. I am very happy you are interested in this question. Your idea is very interesting(using characteristic classes).
Since I asked this questions 2-3 years ago I do not remember the existence of a final answer to this question. I vaguely remember that (maybe) I saw an answer to this question somewhere(may be MO, MSE or RG. But I am not sure at all. May be I am mistaken, so the problem is worth of research.
Regarding embedding in Euclidean space, TS^n is orientable hence can be emmedded in dimension 4n-1. But your suggestion is very interesting:"What is the minimum possible dimension?? very interesting point1 thanks! I think more about it.
I have a remark (not an answer). $TS^n$ is diffeomorphic to the manifold of oriented lines (not necessarily passing through the origin) in $\mathbb{R}^{n+1}$. To see this, let $l$ be an oriented line in $\mathbb{R}^{n+1}$, and let $p \in l$ be the point on $l$ which is the closest to the origin (with respect to the Euclidean distance in $\mathbb{R}^{n+1}$). Let $v$ be a unit tangent direction to $l$, compatible with the orientation of $l$. Then $(v,p)$ define a point on $TS^n$, since $v$ and $p$ are orthogonal. This actually defines a diffeomorphism from the manifold of oriented lines in $\mathbb{R}^{n+1}$ and $TS^n$.
I think a generalization of the Borsuk-Ulam theorem may perhaps show that such a diffeomorphism is impossible. This is an idea.
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