Is the unit tangent bundle of $S^{n}$ a parallelizable manifold. This is motivated by the fact that $TS^{n}$ is parallelizable?
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5$\begingroup$ I thought the tangent bundle $TS^n$ is parallelizable if and only if $n\in\{1;3;7\}$. For $n$ even you won't even find a nowhere vanishing vector field on $S^n$! $\endgroup$– Tommaso CentelegheCommented Nov 5, 2014 at 15:55
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2$\begingroup$ @AliTaghavi, it would probably help to mention the ambiguity that naturally arises here. You might include your qualification in the statement of the question. $\endgroup$– Peter CrooksCommented Nov 5, 2014 at 16:02
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1$\begingroup$ @PeterCrooks I consider $M=$ the unit tangent bundle of $S^{n}$. Is $M$, as a manifold, parallelizable? Note that there is no a primary obstruction since the euler charactristic iz $0=0\times 2$ $\endgroup$– Ali TaghaviCommented Nov 5, 2014 at 16:04
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3$\begingroup$ The unit tangent bundle of the 2-sphere is parallelisable. In fact, every orientable 3-manifold is parallelisable. The latter can be proven by Computing $w_2=0$. $\endgroup$– ThiKuCommented Nov 5, 2014 at 16:11
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2$\begingroup$ The unit tangent bundle of a sphere is usually just called a Stiefel manifold (of 2-frames). $\endgroup$– Ryan BudneyCommented Nov 5, 2014 at 17:39
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1 Answer
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W.Sutherland. A note on the parallelizability of sphere bundles over sphere. J. London Math. Soc. 39 (1964), 55--62.
The answer is yes.
answered Nov 5, 2014 at 18:10
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$\begingroup$ Thank you very much for your answer and the reference. $\endgroup$ Commented Nov 5, 2014 at 20:13