Generalization of the hairy ball theorem. The hairy ball theorem of states that there is no nonvanishing continuous tangent vector field on even dimensional n-spheres.
Can the hairy ball theorem be strengthened to say that there is no continuous tangent vector bundle of dimension 1 on even dimensional n-spheres?
Thanks

I'm interpreting "continuous tangent vector bundle of dimension 1" to mean a rank $1$ subbundle of $TS^{2n}$, i.e., a $1-$dimensional distribution.
In this case, we have the following result (I do not know who first proved it).

If $E\subseteq TS^{2n}$ is a rank $k$ subbundle, then either $k=0$ or $k=2n$.  In other words, $TS^{2n}$ has no non-trivial subbundles.

Proof:  Suppose $E\subseteq TS^{2n}$ is a subbundle.  Since $TS^{2n}$ admits a fiberwise metric, $E$ has a complement $E^\bot$ in the sense that $E\oplus E^\bot \cong TS^{2n}$.
By the Whitney sum formula for the Euler Class, we know $e(TS^{2n}) = e(E)\cup e(E^\bot)$.  Identifying the Euler class with the Euler characteristic, we have $e(TS^{2n}) = 2\in H^{2n}(S^{2n})$, so $e(E)\cup e(E^\bot) = 2$.  Since $H^k(S^{2n}) = 0$ unless $k = 0$ or $k=2n$, the result follows.
