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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 $S^7$? –  Willie Wong Mar 7 '12 at 16:49
    
$S^3$, at least, is parallelizable. All 3-manifolds have trivial tangent bundles. –  Neal Mar 7 '12 at 16:49
    
@WillieWong: Nonvanishing vector fields on $S^{3}$ and $S^{7}$ such that the vector intersects the tangent hyperplane at the same angle throughout, yes. –  deoxygerbe Mar 7 '12 at 17:03
    
Are you considering $S^3$ and $S^7$ as subsets of $\mathbb{R}^4$ and $\mathbb{R}^8$, respectively? –  Neal Mar 7 '12 at 21:12
    
@Neal: I am, yes. –  deoxygerbe Mar 7 '12 at 22:16

1 Answer 1

I think what you want is this: 1) write points on the circle as $z=(x,y)$ or $x+iy$, working in the algebra of complex numbers, then the unit inward normal to the circle at point $z$ is $N=-z$, and the various (constant length) vectors pointing inward at constant angle to $N$ are $N+aiz=(-x-ay,-y+ax)$ for any real constant $a$, or positive multiples of that. What you called right and left of the inner normal correspond to $a$ positive and negative respectively.

2) Write points on the 3-sphere as $q=(w,x,y,z)$ or $w+ix+jy+kz$, working in the algebra of quaternions, then the unit inward normal to the 3-sphere at point $q$ is $N=-q$, and the constant length vectors pointing inward at constant angle to $N$ are $N+(ai+bj+ck)q$ for any real constants $a$, $b$, $c$, or positive multiples of that.

3) For the 7-sphere use the same type of construction with the octonian algebra.

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