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On $S^{1}$ there are three choices for a vector pointed inward to the same angle at the tangent line at every point. The vector is either in the opposite direction as the normal, or to the left of it, or to the right of it. (clockwise, counterclockwise, and all inward)

I can skip $S^{2}$ because of the hairy ball theorem. What about $S^{3}$ and $S^{7}$? I know that they're both uncombable.

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    I don't quite understand your question. What do you actually mean by vortices? Are you asking about how to classify non-vanishing vector fields on $S^3$ and $S^7$?2012-03-07
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    $S^3$, at least, is parallelizable. All 3-manifolds have trivial tangent bundles.2012-03-07
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    @WillieWong: Nonvanishing vector fields on $S^{3}$ and $S^{7}$ such that the vector intersects the tangent hyperplane at the same angle throughout, yes.2012-03-07
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    Are you considering $S^3$ and $S^7$ as subsets of $\mathbb{R}^4$ and $\mathbb{R}^8$, respectively?2012-03-07
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    @Neal: I am, yes.2012-03-07
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    @deoxygerbe: every vector field on $S^k$ (as a embedded submanifold of $R^{k+1}$ can be written as $v = v_n N + v_p$ where $v_n$ is the normal component to $S^k$ and $v_p$ is a vector field along $S^k$ that is tangent to it. By rescaling $v_p$ (multiplying it by a function), you see that your question is **equivalent** to classifying all nonvanishing vector fields _along_ (tangent to) the spheres.2012-03-08

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