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We have known how many linear independent vector fields can be constructed on $S^n$:https://en.wikipedia.org/wiki/Vector_fields_on_spheres

So how many linear independent vector fields can be constructed on a general other manifold with $\chi(M)=0$? Is there any progress on this area?

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    $\begingroup$ There's a bound coming from computing Stiefel-Whitney classes. It should imply that in general you can only get one. $\endgroup$
    – Qiaochu Yuan
    Commented Apr 6, 2016 at 5:34
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    $\begingroup$ This is called the span of the manifold $M$, and is well studied since the 1960s. $\endgroup$
    – Mark Grant
    Commented Apr 6, 2016 at 8:36

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